An Interior Trajectory Simulation of the Gas- steam ... - Semantic Scholar
JOURNAL OF COMPUTERS, VOL. 8, NO. 5, MAY 2013
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An Interior Trajectory Simulation of the Gassteam Missile Ejection Yongquan Liu1,2, Anmin Xi1 1.University of Science and Technology Beijing, Beijing, China; 2. Inner Mongolia University of Science and Technology, Baotou, China Email: [email protected]
Abstract—An interior trajectory simulation of the canister launched missiles has been developed with the help of FLUENT. One of the special features of this simulation is the method by which the coupled three-phase problem is reduced to solving the fluid equations only. Several new source terms describing the influence of droplets and particles on gas have been created in the present case and added to the motion equations and energy equations of the gas. According to the simulation, the almost uniform acceleration occurs 0.25s after startup and indicates that the missile is shockless during ejection. The calculation has been done efficiently in FLUENT after setting all the required parameters. Several figures obtained from the simulation show the velocity and pressure of the flow field in the reservoir. The distribution curves of the velocity and acceleration show that the simulation and the test data of the experiments are in good agreement with each other. This model satisfies the requirements of the missile ejection and can be used to analyze the similar launch procedures. Index Terms—interior ejection
ballistic
simulation,
gas-steam
I. INTRODUCTION Many new methods have been presented with the development of the launching technology of the missile. Two general categories of canister launched techniques are recognized, self eject launch and gas eject launch. Each specific technique has its own particular benefits and problems. One of the most promising gas eject techniques is the use of a hot gas generator to provide the force to drive the missile out of the canister. In this procedure, a solid propellant gas generator is ignited in the closed volume(reservoir) below the missile in the canister and the resultant pressurization forces the missile out. In order to avoid damage to the bottom of the missile and the walls of the canister, cooling water is provided to mix with the gas before they reach the closed volume, which is known as the gas-steam eject launch method. Shouzhen Rui et al. [1] studied several ejection method such as gas-steam ejection, gas ejection and compressed air ejection with respect to ejection power system, structure, change of pressure and acceleration, temperature, velocity and displacement and launching Corresponding author: Yongquan Liu, male, Ph.D, email: [email protected].
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time. They found that the gas-steam method did better in all respects. In the paper of C. T. Edquist and G. Romine [2], an analysis procedure was developed for studying the launch acceleration pulse of a gas-steam ejection missile. Particular attention was given to modeling the loss mechanisms. The inclusion of the heat loss was the primary difference between this paper and previous formulations. In addition, both perfect and chemical equilibrium gas calculations were considered. Eight years later, C. T. Edquist [3] developed a gas dynamic mode1 of the process involved in launching the Small ICBM(Intercontinental Ballistic Missile) from a canister. He thoroughly analyzed the launch phase to accurately predict the maximum load occurring during the launch. It was related with the HGG(hot gas generator) chamber pressure, the propellant and coolant properties, eject system flow areas, and missile and launch tube physical characteristics. These two papers of C. T. Edquist were based on engineering thermodynamics with many assumptions. These equations were very important and fundamental, but short of details. In order to obtain more of the flow field of the whole system, many scientists made various attempts. Some of them focused on the parts of the flow field while some were interested in the phases of the flow. Yu. N. Sadkov [4] investigated the flow of free gas jets from a sonic nozzle, which is the most important component of the ejection system, using the axisymmetric Euler equation model. Similarly, Vincent Lijo et al. [5] presented a numerical investigation of transient flows in an axisymmetric nozzle. Both papers showed a qualitative good agreement with the experimental results. D. Albagli and Y. Levy [6] and M. Sommerfeld[7] experimentally studied the two phase flow to examine the influence of the dispersed particles on the gas flow and the structure of the imbedded shock waves. Qiang Qi et al. [8] and Yanhui Tang et al. [9] simulated the interior trajectory using three-phase flow method, updating dynamic grid and turbulent model. The errors of their simulations were small. However, they all had to deal with the three-phase flow model appearing in a traditional form in a complex way. In this paper, we propose a new method by which the three-phase model can be simplified and this makes it easier to simulate the interior trajectory of gas-steam missile ejection. In addition, the entrance of the mixture inclines to the canister at an angle of 45 to prevent the gas-steam jet
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JOURNAL OF COMPUTERS, VOL. 8, NO. 5, MAY 2013
from impinging on the missile and the sidewall directly. Figure 1 is the illustration of the system. launch canister
gas generator and cooling water device
tiny particle and the tiny droplet can keep up with the gas flow to some extent, but they can't catch up when the gas is fluctuating. Therefore, ui − u pi = ui′ and ui − u di = ui′ . According to the law of action and reaction, the reaction forces of Fpi and Fdi act on the surrounding gas in the opposite direction and they are always resistance,
missile lateral support pads
Fpi′ = C D
1 2 1 πd p ρ ui′ui′ 4 2
(3)
reservoir
Fdi′ = C D′
1 2 1 πd d ρ ui′ui′ 4 2
(4)
In (3) and (4), Fpi′ and Fdi′ are the reaction forces of
Figure 1. Canister launched missile schematic
Fpi and Fdi .
With setting unsteady flow, dynamic mesh(layering) and UDFs(User Defined Functions) in FLUENT, the most widely used CFD application software, a simulation has been done to obtain the state of the flow field in the canister during the ejecting of the missile.
We assume that the particles and the droplets are uniformly distributed and they are located at the center of the fluid cube. Now change the form of (3) and (4), so that we can add the influence of Fpi′ and Fdi′ to Navierstokes equation of the gas.
II. CALCULATIONS
ρ
A. Equations The gas flow rushing into the canister is the threephase flow, including hot gas, droplets coming from the cooling water and particles of the explosives. The main influence of fluid on particles is the resisting force Fpi , Fpi = C D Ap
1 ρ (ui − u pi ) ui − u pi 2
(1)
1 2 πd p ( d p is the mean diameter of 4 particles and known by measurement. In the present case, d p =20μm.), C D is a coefficient related to Re and Ma, ρ
where Ap =
ρ
Fdi = C D′ Ad
1 ρ (ui − u di ) ui − u di 2
(2)
1 where Ad = πd d2 ( d d is the mean diameter of droplets, 4 d d =15μm.), C D′ is a coefficient related to Re and Ma, and u di is the velocity of the droplet. Considering the form of the equations above, we can simplify the three-phase flow equations in the following way so as to avoid solving the equations of the particles and droplets. In (1), ui = ui + ui′ . Since the particles and the droplets are very tiny, let u pi = ui , u di = ui . This means that the
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3ϕ p
1 2
ρ ui′ui′
(5)
Fdi′ 3ϕ d 1 = C D′ ρ ui′ui′ mcube 2d d 2
(6)
mcube
= CD
2d p
In (5) and (6), ϕ p and ϕ d are the volume fraction of the particles and droplets. After we split each property into mean plus fluctuating variables and take the time mean of the equation (5) and (6),
ρ
is the density of the gas, ui is the velocity of the gas and u pi is the velocity of the particle. The mean diameter of the droplets of the cooling water is 15μm according to statistics, so we can treat the droplets as particles and use the same equation to describe the resisting force applied by gas,
Fpi′
ρ
Fpi′
3ϕ p
1 2
ρ ui′ui′
(7)
Fdi′ 3ϕ d 1 ρ ui′ui′ = C D′ mcube 2d d 2
(8)
mcube
= CD
2d p
1 ui′ui′ = k . 2 Now the term concerning Fpi′ and Fdi′ consist of the
In the above two equations,
elements which are have nothing to do with the motion of the particles and droplets in form. In this case, the particle equations and the droplet equations of the three-phase flow can be discarded. It's enough for us to solve the gas equations only. The complete motion equation is: ⎞ ∂ui ∂u ∂p ∂ ⎛⎜ ∂ui + ρu j i = − + − ρ ui′u ′j ⎟ + μ ⎟ ∂t ∂x j ∂xi ∂x j ⎜⎝ ∂x j ⎠ 3ϕ p 1 3ϕ d 1 CD ρ ui′ui′ + C D′ ρ ui′ui′ 2d p 2 2d d 2
ρ
The energy equation is in the same situation.
(9)
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Navier-stokes equation multiplied by ui is the energy equation. ∂ ⎡ρ ∂ ⎡ρ ∂p ⎤ ⎤ ui ui + ui′ui′ ⎥ + u j ui ui + ui′ui′ ⎥ + u j + ⎢ ⎢ ∂t ⎣ 2 ∂x j ⎣ 2 ∂x j ⎦ ⎦ ∂ ⎡ ⎛⎜ ∂ui ∂u j ⎞⎟⎤ ρ ∂ ⎛ ⎞ + u ′j ⎜ p′ + ui′ui′ ⎟ = μ ⎢ui ⎥+ ∂x j ⎣⎢ ⎜⎝ ∂x j ∂xi ⎟⎠⎦⎥ 2 ∂x j ⎝ ⎠ ∂ ∂ ⎛⎜ ∂ui′ ∂u ′j ⎞⎟ + − ui − ρ ui′u ′j + μ ui′ ∂x j ∂x j ⎜⎝ ∂x j ∂xi ⎟⎠ ⎛ ∂u ∂u ⎞ ∂u ∂u ′ ⎛ ∂u ′ ∂u ′ ⎞ μ ⎜⎜ i + j ⎟⎟ i − μ i ⎜⎜ i + j ⎟⎟ + ∂x j ⎝ ∂x j ∂xi ⎠ ⎝ ∂x j ∂xi ⎠ ∂x j 3ϕ p 3ϕ 1 1 ρui ui′ui′ + C D p ρ ui′(ui′ui′)′ + CD 2d p 2 2d p 2 3ϕ d 3ϕ 1 1 C D′ ρui ui′ui′ + C D′ d ρ ui′(ui′ui′)′ 2d d 2 2d d 2 (10) ′ ′ ′ ′ In (10), the terms containing ui (ui ui ) can be ignored. The above discussion and equations apply to the most part of the fluid(gas) field in the canister. However, the gas flow near the wall of the canister may be a little different from the others due to the elastic moving particles. The particles in the gas flow will bounce off the canister wall when they hit the bottom or the sidewall. Because the jetflow will hit the wall several times, we have to study the extra influence of the particles on the gas flow near the wall of the canister.
(
)
[(
(
)
)]
c
b
a (a)
(b)
Due to the inviscid and elastic property, the particle will “deviate” from the “track” along the boundary, which results in the extra force exerted on the gas. Fig.2 describes the motion of particles near the wall. We still use the equation (3) to describe the influence of this extra force Fei′ on the gas. 1 ρu n′ u n′ 2
(11)
3ϕ 3ϕ Fei′ 1 1 1 = C D p ρ u n′ u n′ ≈ C D p ρ • • ui′ui′ mcube 2d p 2 2d p 3 2
[(
)
(
(12)
Because the elastic collision, the ui − u pi term in
)
)]
⎞ ∂k ⎤ ⎟⎟ ⎥ + Gk − ρε ⎠ ∂x j ⎦⎥ (15) ⎡ ⎤ μ ⎞ ∂ε ∂( ρε ) ∂ ( ρεui ) ∂ ⎛ = + ⎢⎜⎜ μ + t ⎟⎟ ⎥ σ ε ⎠ ∂x j ⎥⎦ ∂t ∂xi ∂x j ⎢⎣⎝ (16)
∂ ∂ ( ρk ) ∂ ( ρkui ) + = ∂x j ∂t ∂xi
⎡⎛ μ ⎢⎜⎜ μ + t σ k ⎣⎢⎝
ε2
k + νε where σ k = 1.0 , σ ε = 1.2 , C2 = 1.9 ,
⎛ η ⎞ ⎟, C1 = max⎜⎜ 0.43, η + 5 ⎟⎠ ⎝
η = (2 Eij • Eij )
1
2
k
ε
, Eij =
1 ⎛⎜ ∂ui ∂u j + 2 ⎜⎝ ∂x j ∂xi
⎞ ⎟. ⎟ ⎠
In (16), μt is given by
k2
ε
In (17), the parameter Cμ can be defined as
Cμ =
resisting force equation will be enhanced. Add this term to the motion and energy equations of the gas near the wall.
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(
μt = ρCμ
Take the same steps as before to deal with equation (11).
ρ
(13) The corresponding energy equation is: ∂ ⎡ρ ∂ ⎡ρ ∂p ⎤ ⎤ ui ui + ui′ui′ ⎥ + u j ui ui + ui′ui′ ⎥ + u j + ⎢ ⎢ ∂t ⎣ 2 ∂x j ⎣ 2 ∂x j ⎦ ⎦ ρ ∂ ∂ ⎡ ⎛⎜ ∂ui ∂u j ⎞⎟⎤ ⎛ ⎞ u ′j ⎜ p′ + ui′ui′ ⎟ = μ + ⎢ui ⎥+ 2 ∂x j ⎝ ∂x j ⎢⎣ ⎜⎝ ∂x j ∂xi ⎟⎠⎥⎦ ⎠ ∂ ⎛⎜ ∂ui′ ∂u ′j ⎞⎟ ∂ − ui − ρ ui′u ′j + μ ui′ + ∂x j ⎜⎝ ∂x j ∂xi ⎟⎠ ∂x j ⎛ ∂u ∂u ⎞ ∂u ∂u ′ ⎛ ∂u ′ ∂u ′ ⎞ μ ⎜⎜ i + j ⎟⎟ i − μ i ⎜⎜ i + j ⎟⎟ + ∂x j ⎝ ∂x j ∂xi ⎠ ⎝ ∂x j ∂xi ⎠ ∂x j 3 3ϕ ϕ 4 1 4 1 C D p ρui ui′ui′ + C D p ρ ui′(ui′ui′)′ + 3 2d p 2 3 2d p 2 3ϕ d 3ϕ 1 1 C D′ ρui ui′ui′ + C D′ d ρ ui′(ui′ui′)′ 2d d 2 2d d 2 (14) It is acceptable to choose Realizable k −ε turbulence model [10-17] in calculation in FLUENT. The Realizable k − ε equations are as follows:
+ ρC1 Eij ε − ρC2
Figure 2. Particle motion near the wall
Fei′ = C D Ap
The motion equation near the wall is: ⎞ ∂u ∂u ∂p ∂ ⎛⎜ ∂ui + − ρ ui′u ′j ⎟ + ρ i + ρu j i = − μ ⎟ ⎜ ∂t ∂x j ∂xi ∂x j ⎝ ∂x j ⎠ 3ϕ p 1 3ϕ d 1 4 CD ρ ui′ui′ + C D′ ρ ui′ui′ 3 2d p 2 2d d 2
where
1 A0 + ASU *k / ε
(17)
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A0 = 4.0 ⎫ ⎪ AS = 6 cos φ ⎪ 1 φ = cos −1 ( 6W ) ⎪ ⎪ 3 ⎪ Eij E jk Ekj ⎪ W= ( Eij Eij )1 2 ⎪ ⎬ 1 ⎛⎜ ∂ui ∂u j ⎞⎟ ⎪ Eij = + 2 ⎜⎝ ∂x j ∂xi ⎟⎠ ⎪ ~ ~ ⎪ U * = Eij Eij + Ω ij Ω ij ⎪ ⎪ ~ Ω ij = Ω ij − 2ε ijk ωk ⎪ Ω ij = Ωij − ε ijk ωk ⎪⎭ Parameter Ω is a time mean speed tensor relative to the selected reference frame whose angular velocity is ωk . Realizable k − ε model used to describe the fully developed turbulence is applicable to most part of the flow field in the canister. We have to use different model to solve the flow field near the wall of the canister because the low Reynolds number, turbulence which is not fully developed and viscosity dominate the near-wall region [18]. We choose the low Reynolds number k − ε model proposed by Jones and Launder [19-22] to describe the mentioned region. μ ⎞ ∂k ⎤ ∂ ( ρk ) ∂ ( ρkui ) ∂ ⎡⎛ + = ⎥ ⎢⎜⎜ μ + t ⎟⎟ σ k ⎠ ∂x j ⎥⎦ ∂t ∂xi ∂x j ⎢⎣⎝ (18) 2 ⎛ ∂k 1 2 ⎞ ⎟ + Gk − ρε − 2 μ ⎜⎜ ⎟ ⎝ ∂n ⎠
∂ ( ρε ) ∂ ( ρεui ) ∂ + = ∂t ∂xi ∂x j
⎡⎛ μ ⎢⎜⎜ μ + t σε ⎣⎢⎝
ε
k2
ε
, Cμ = 0.09 , σ k = 1.0
⎛ ∂u ∂u j ⎞ ∂ui ⎟ , f1 = 1.0 , Gk = μt ⎜ i + ⎜ ∂x j ∂xi ⎟ ∂x j ⎠ ⎝ f 2 = 1.0 − 0.3 exp(− Re t2 ) , f μ = exp(−2.5 /(1 + Re t / 50)) , and
Re t = ρk 2 /(ηε ) . The equations we finally need in the present case when we deal with the gas flow in most part of the canister are
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B. Models The physical model of the system can be simplified as follows:
⎞ ∂ε ⎤ ⎟⎟ ⎥ ⎠ ∂x j ⎦⎥
(19) 2 2 μμt ⎛ ∂ 2u ⎞ C1ε ε ε2 ⎜ ⎟ + Gk f1 − C2ε ρ f2 + ρ ⎜⎝ ∂n 2 ⎟⎠ k k where n represents the normal coordinate of the wall, u is the speed parallel to the canister wall, k2 μt = Cμ f μ ρ , C1ε = 1.44 , C2ε = 1.92 ,
σ ε = 1.3 , μt = ρCμ
⎧ ∂ui ⎪ ∂x = 0 ⎪ i ⎪ (9) ⎨ (10) ⎪ ⎪ (15) ⎪ ⎩ (16) The equations near the wall are ⎧ ∂ui ⎪ ∂x = 0 ⎪ i ⎪ (13) ⎨ (14) ⎪ ⎪ (18) ⎪ ⎩ (19)
Figure 3. Meshed physical model
In Fig. 3, the top surface of the model represents the bottom of the missile which is expected to be raised up by the pressure of the gas in the reservoir. The missile will not move until the pressure in the reservoir reaches the critical value. The time this procedure takes varies depending on the inlet flow rate or the inlet pressure. Then the missile moves upwards driven by the gas which is forced into the reservoir continuously from the inlet.
C. Calculations and Conclusions The values of the inlet pressure which are derived from test data, the parameters of the moving zone(dynamic mesh) [23] which is described by the DEFINE_CG_MOTION() macro and the source terms we added to the governing equations are defined by the UDFs(User Defined Functions) [24,25].Below is an excerpt from the moving zone UDF. DEFINE_CG_MOTION(piston,dt,vel,omega,time,dti me) { Thread *t; face_t f;
JOURNAL OF COMPUTERS, VOL. 8, NO. 5, MAY 2013
Message("time=%f,z_vel=%f,force=%f\n",time,v_prev ,force); vel[2]=v_prev; } ... The flow field conditions of the x=0 section are shown in following figures. Fig. 4 and Fig. 5 show the total pressure at 0.07s and 0.36s respectively in the reservoir, while the velocity field under the missile at 0.36s is given in Fig. 6.
makes the pressure drop a little. With the continuous gassteam injection, the stable and enough pressure near the missile is maintained until the missile is pushed out of the launcher.
Figure 6. Velocity field at 0.36s
The gas is injected into the reservoir at a very high speed. Then it rushes towards the bottom of the missile with several eddies in the middle. When it approaches the bottom, its field of velocity becomes uniform. veloctiy
calculations test data
40 velocity (m/s)
real NV_VEC(A); real force,dv; NV_S(vel,=,0.0); NV_S(omega,=,0.0); if (!Data_Valid_P()) return; t=DT_THREAD(dt); force=0.0; begin_f_loop(f,t) { F_AREA(A,f,t); force+=F_P(f,t)*NV_MAG(A); } end_f_loop(f,t) ....
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30 20 10 0 0
0.2
0.4 time (s)
0.6
0.8
Figure 7. Velocity distribution curves
Figure 4. Pressure field at 0.07s
Fig. 7 shows the velocity distribution curves of the missile. According to the calculations(the blue curve), it takes 0.10s for the pressure in the reservoir to exceed the sum of the gravity of the missile and the resisting force which are calculated in the corresponding UDF. After 0.25 seconds, the velocity curve becomes almost linear, which indicates the uniform acceleration and shows good agreement with the pressure variation near the missile bottom. When the missile departs from the canister at 0.65s, its velocity is 37.2m/s. This curve is in conformity with the test data acquired from the experiments(the pink). calculations test data
Figure 5. Pressure field at 0.36s
The pressure near the inlet tops the whole pressure field in the reservoir. At an early stage of the ejection, the pressure in the reservoir increases rapidly because of the small reservoir volume and the missile at rest. Then the gas in the reservoir expands along with the moving missile, which © 2013 ACADEMY PUBLISHER
acceleration (m/s2)
acceleration 120 100 80 60 40 20 0 0
0.2
0.4 time (s)
0.6
Figure 8. Acceleration distribution curves
0.8
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JOURNAL OF COMPUTERS, VOL. 8, NO. 5, MAY 2013
Fig. 8 shows the acceleration-time histories of the missile. The calculations(the blue curve) indicate that the peak value of the acceleration(95.5m/s2) occurs at 0.31s.Then it decreases a little. The final value is 62.8m/s2 with time at 0.63s.These are very close to the values obtained from the experiments(the pink). Both the peak value of the calculations and the max. test data are acceptable according to the material of the missile bottom. III. CONCLUDING REMARKS A simulation has been developed for analyzing the gas dynamics of canister launched missiles. Unique to the technique is the acceleration or the pressure of the missile, which is uniform during the most ejection time. Inclusion of these measures, such as dynamic mesh, k − ε model, and so on, allows a reliable simulation of the ejection. Results of the simulation show that it is in good agreement with test data. This model can be used to analyze the similar launch procedures. REFERENCES [1] Shouzhen Rui and Yuming Xing, “Comparative studies of interior ballistic performance among several missile eject power systems,” Journal of Beijing University of Aeronautics and Astronautics, Vol.35, pp.766–770, June 2009. [2] C. T. Edquist and G. L. Romine, “Canister Gas Dynamics of Gas Generator Launched Missiles,” AIAA-80-1186. [3] C. T. Edquist, “Prediction of the Launch Pulse for Gas Generator Launched Missiles,” AIAA-88-3290. [4] Yu. N. Sadkov, “Numerical simulation of supersonic axisymmetric gas jets,” Computational Mathematics and Modeling, vol. 15, pp. 344–349, April 2004. [5] Vincent Lijo, Heuy Dong Kim, Toshiaki Setoguchi, and Shigeru Matsuo, “Numerical simulation of transient flows in a rocket propulsion nozzle,” International Journal of Heat and Fluid Flow, vol. 31, pp. 409–417, June 2010. [6] D. Albagli and Y. Levy, “Experimental study on confined two-phase jets,” J. Thermophysics, vol. 5, pp. 387–393, July 1991. [7] M. Sommerfeld, “The structure of particle-laden, underexpanded free jets,” Shock Waves, vol. 3, pp. 299– 311, March 1994. [8] Qiang Qi, Qinggui Chen, Yuan Zhou, Haiyang Wang, and Hongmei Zhou, “Submarine-Launched Cruise Missile Ejecting Launch Simulation and Research,” 2011 International Conference on Electronic & Mechanical Engineering and Information Technology, pp.4542–4545, August 2011. [9] Yanhui Tang, Yaoguo Xing, and Chunlong Zhang, “Working Process Simulation of Gas and Steam Launching System,” Journal of Naval Aeronautical and Astronautical University, Vol.24, pp.431–434, July 2009. [10] Xiaochao Zhai, Yajun Chen, and Yi Jiang, “Using dynamic grid to simulate interior trajectory of the launching container,” Modern Defence Technology, Vol.34, pp.24–28, April 2006. [11] B. E. Launder and D. B. Spalding, “The numerical computation of turbulent flows,” Computer Methods in Applied Mechanics and Engineering, pp.269-289, March 1974.
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[12] Shiwu Fang, Xueguang Zhang, and Haiying Qian, “Research on the Valid Powder Energy Coefficient for Canister Type Missile Ejection Launcher,” Missiles and Space Vehicles, pp. 18–24, March 2004. [13] Yi Jiang, Jiguang Hao, and Debin Fu, “3D Unsteady Numerical Simulation of Missile Launching,” ACTA ARMAMENTARII., vol. 29, pp. 911–915, August 2008. [14] B. Raverdy, I. Mary, P. Sagaut, and N. Liamis, “Highresolution large-eddy simulation of flow around lowpressure turbine blade,” AIAA Journal, pp.390–397, March 2003. [15] I. Demirdzic and M. Peric, “Finite volume methods for prediction of fluid flow in arbitrarily shaped domains with moving boundaries,” Int. J. Num. Meth. Fluids, pp.771– 790, October 1990. [16] M. Ishii and K. Mishima, “Two-fluid model and hydrodynamic constitutive relation,” Nuclear Engineering and Design, Vol.82, pp.107–126, March 1984. [17] Hassan, “Mesh generation and adaptivity for the solution of compressible viscous high speed flow,” Int. J. Numerical Methods Engineering, Vol.38, pp.1123–1148, Dec. 1995. [18] R. M. C. So, Y. G. Lai, and H. S. Zhang, “Second-order near-wall turbulence closures: a review,” AIAA Journal, Vol.29, pp.1819–1835, April 1991. [19] C. G. Speziale, “Analytical methods for the development of Reynolds-stress closures in turbulence,” Ann. Rev. Fluid Mech., Vol.23, pp.107–157, March 1991. [20] B. E. Launder, “Second-moment closure and its use in modeling turbulent industrial flows,” International Journal for Numerical Methods in Fluids, Vol. 9, pp.963–985, Sep.1989. [21] B. E. Launder, “Second-moment closure: present and future,” J. Heat Fluid Flow, Vol.10, pp.282–300, April 1989. [22] W. P. Jones and B. E. Launder, “The calculation of lowReynolds-number phenomena with a two-equation model of turbulence,” Int. J. Heat Mass Transfer, Vol.16, pp.1119–1130, June 1983. [23] E. J. Probert, O. Hassan, K. Morgan, and J. Peraire, “An adaptive finite element method for transient compressible flows with moving boundaries,” Int. J. Numerical Methods Engineering, Vol.32, pp.1145–1159, Nov.1991. [24] Fluent Inc., FLUENT User’s Guide, Fluent Inc..USA, 2003. [25] K. F. C. Yiu, D.M.Greaves, S. Cruz, A. Saalehi, and A. G. L. Borthwick, “Quadtree grid generation: Information handling, boundary fitting and CFD applications,” Comput. Fluids, Vol.25, pp.759–769, July 1996.
Yongquan Liu Date of birth: 08.20.1980, male, PH.D in University of Science and Technology Beijing, Research Fields: fluid mechanics, mechanical engineering.
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An Interior Trajectory Simulation of the Gassteam Missile Ejection Yongquan Liu1,2, Anmin Xi1 1.University of Science and Technology Beijing, Beijing, China; 2. Inner Mongolia University of Science and Technology, Baotou, China Email: [email protected]
Abstract—An interior trajectory simulation of the canister launched missiles has been developed with the help of FLUENT. One of the special features of this simulation is the method by which the coupled three-phase problem is reduced to solving the fluid equations only. Several new source terms describing the influence of droplets and particles on gas have been created in the present case and added to the motion equations and energy equations of the gas. According to the simulation, the almost uniform acceleration occurs 0.25s after startup and indicates that the missile is shockless during ejection. The calculation has been done efficiently in FLUENT after setting all the required parameters. Several figures obtained from the simulation show the velocity and pressure of the flow field in the reservoir. The distribution curves of the velocity and acceleration show that the simulation and the test data of the experiments are in good agreement with each other. This model satisfies the requirements of the missile ejection and can be used to analyze the similar launch procedures. Index Terms—interior ejection
ballistic
simulation,
gas-steam
I. INTRODUCTION Many new methods have been presented with the development of the launching technology of the missile. Two general categories of canister launched techniques are recognized, self eject launch and gas eject launch. Each specific technique has its own particular benefits and problems. One of the most promising gas eject techniques is the use of a hot gas generator to provide the force to drive the missile out of the canister. In this procedure, a solid propellant gas generator is ignited in the closed volume(reservoir) below the missile in the canister and the resultant pressurization forces the missile out. In order to avoid damage to the bottom of the missile and the walls of the canister, cooling water is provided to mix with the gas before they reach the closed volume, which is known as the gas-steam eject launch method. Shouzhen Rui et al. [1] studied several ejection method such as gas-steam ejection, gas ejection and compressed air ejection with respect to ejection power system, structure, change of pressure and acceleration, temperature, velocity and displacement and launching Corresponding author: Yongquan Liu, male, Ph.D, email: [email protected].
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time. They found that the gas-steam method did better in all respects. In the paper of C. T. Edquist and G. Romine [2], an analysis procedure was developed for studying the launch acceleration pulse of a gas-steam ejection missile. Particular attention was given to modeling the loss mechanisms. The inclusion of the heat loss was the primary difference between this paper and previous formulations. In addition, both perfect and chemical equilibrium gas calculations were considered. Eight years later, C. T. Edquist [3] developed a gas dynamic mode1 of the process involved in launching the Small ICBM(Intercontinental Ballistic Missile) from a canister. He thoroughly analyzed the launch phase to accurately predict the maximum load occurring during the launch. It was related with the HGG(hot gas generator) chamber pressure, the propellant and coolant properties, eject system flow areas, and missile and launch tube physical characteristics. These two papers of C. T. Edquist were based on engineering thermodynamics with many assumptions. These equations were very important and fundamental, but short of details. In order to obtain more of the flow field of the whole system, many scientists made various attempts. Some of them focused on the parts of the flow field while some were interested in the phases of the flow. Yu. N. Sadkov [4] investigated the flow of free gas jets from a sonic nozzle, which is the most important component of the ejection system, using the axisymmetric Euler equation model. Similarly, Vincent Lijo et al. [5] presented a numerical investigation of transient flows in an axisymmetric nozzle. Both papers showed a qualitative good agreement with the experimental results. D. Albagli and Y. Levy [6] and M. Sommerfeld[7] experimentally studied the two phase flow to examine the influence of the dispersed particles on the gas flow and the structure of the imbedded shock waves. Qiang Qi et al. [8] and Yanhui Tang et al. [9] simulated the interior trajectory using three-phase flow method, updating dynamic grid and turbulent model. The errors of their simulations were small. However, they all had to deal with the three-phase flow model appearing in a traditional form in a complex way. In this paper, we propose a new method by which the three-phase model can be simplified and this makes it easier to simulate the interior trajectory of gas-steam missile ejection. In addition, the entrance of the mixture inclines to the canister at an angle of 45 to prevent the gas-steam jet
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from impinging on the missile and the sidewall directly. Figure 1 is the illustration of the system. launch canister
gas generator and cooling water device
tiny particle and the tiny droplet can keep up with the gas flow to some extent, but they can't catch up when the gas is fluctuating. Therefore, ui − u pi = ui′ and ui − u di = ui′ . According to the law of action and reaction, the reaction forces of Fpi and Fdi act on the surrounding gas in the opposite direction and they are always resistance,
missile lateral support pads
Fpi′ = C D
1 2 1 πd p ρ ui′ui′ 4 2
(3)
reservoir
Fdi′ = C D′
1 2 1 πd d ρ ui′ui′ 4 2
(4)
In (3) and (4), Fpi′ and Fdi′ are the reaction forces of
Figure 1. Canister launched missile schematic
Fpi and Fdi .
With setting unsteady flow, dynamic mesh(layering) and UDFs(User Defined Functions) in FLUENT, the most widely used CFD application software, a simulation has been done to obtain the state of the flow field in the canister during the ejecting of the missile.
We assume that the particles and the droplets are uniformly distributed and they are located at the center of the fluid cube. Now change the form of (3) and (4), so that we can add the influence of Fpi′ and Fdi′ to Navierstokes equation of the gas.
II. CALCULATIONS
ρ
A. Equations The gas flow rushing into the canister is the threephase flow, including hot gas, droplets coming from the cooling water and particles of the explosives. The main influence of fluid on particles is the resisting force Fpi , Fpi = C D Ap
1 ρ (ui − u pi ) ui − u pi 2
(1)
1 2 πd p ( d p is the mean diameter of 4 particles and known by measurement. In the present case, d p =20μm.), C D is a coefficient related to Re and Ma, ρ
where Ap =
ρ
Fdi = C D′ Ad
1 ρ (ui − u di ) ui − u di 2
(2)
1 where Ad = πd d2 ( d d is the mean diameter of droplets, 4 d d =15μm.), C D′ is a coefficient related to Re and Ma, and u di is the velocity of the droplet. Considering the form of the equations above, we can simplify the three-phase flow equations in the following way so as to avoid solving the equations of the particles and droplets. In (1), ui = ui + ui′ . Since the particles and the droplets are very tiny, let u pi = ui , u di = ui . This means that the
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3ϕ p
1 2
ρ ui′ui′
(5)
Fdi′ 3ϕ d 1 = C D′ ρ ui′ui′ mcube 2d d 2
(6)
mcube
= CD
2d p
In (5) and (6), ϕ p and ϕ d are the volume fraction of the particles and droplets. After we split each property into mean plus fluctuating variables and take the time mean of the equation (5) and (6),
ρ
is the density of the gas, ui is the velocity of the gas and u pi is the velocity of the particle. The mean diameter of the droplets of the cooling water is 15μm according to statistics, so we can treat the droplets as particles and use the same equation to describe the resisting force applied by gas,
Fpi′
ρ
Fpi′
3ϕ p
1 2
ρ ui′ui′
(7)
Fdi′ 3ϕ d 1 ρ ui′ui′ = C D′ mcube 2d d 2
(8)
mcube
= CD
2d p
1 ui′ui′ = k . 2 Now the term concerning Fpi′ and Fdi′ consist of the
In the above two equations,
elements which are have nothing to do with the motion of the particles and droplets in form. In this case, the particle equations and the droplet equations of the three-phase flow can be discarded. It's enough for us to solve the gas equations only. The complete motion equation is: ⎞ ∂ui ∂u ∂p ∂ ⎛⎜ ∂ui + ρu j i = − + − ρ ui′u ′j ⎟ + μ ⎟ ∂t ∂x j ∂xi ∂x j ⎜⎝ ∂x j ⎠ 3ϕ p 1 3ϕ d 1 CD ρ ui′ui′ + C D′ ρ ui′ui′ 2d p 2 2d d 2
ρ
The energy equation is in the same situation.
(9)
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Navier-stokes equation multiplied by ui is the energy equation. ∂ ⎡ρ ∂ ⎡ρ ∂p ⎤ ⎤ ui ui + ui′ui′ ⎥ + u j ui ui + ui′ui′ ⎥ + u j + ⎢ ⎢ ∂t ⎣ 2 ∂x j ⎣ 2 ∂x j ⎦ ⎦ ∂ ⎡ ⎛⎜ ∂ui ∂u j ⎞⎟⎤ ρ ∂ ⎛ ⎞ + u ′j ⎜ p′ + ui′ui′ ⎟ = μ ⎢ui ⎥+ ∂x j ⎣⎢ ⎜⎝ ∂x j ∂xi ⎟⎠⎦⎥ 2 ∂x j ⎝ ⎠ ∂ ∂ ⎛⎜ ∂ui′ ∂u ′j ⎞⎟ + − ui − ρ ui′u ′j + μ ui′ ∂x j ∂x j ⎜⎝ ∂x j ∂xi ⎟⎠ ⎛ ∂u ∂u ⎞ ∂u ∂u ′ ⎛ ∂u ′ ∂u ′ ⎞ μ ⎜⎜ i + j ⎟⎟ i − μ i ⎜⎜ i + j ⎟⎟ + ∂x j ⎝ ∂x j ∂xi ⎠ ⎝ ∂x j ∂xi ⎠ ∂x j 3ϕ p 3ϕ 1 1 ρui ui′ui′ + C D p ρ ui′(ui′ui′)′ + CD 2d p 2 2d p 2 3ϕ d 3ϕ 1 1 C D′ ρui ui′ui′ + C D′ d ρ ui′(ui′ui′)′ 2d d 2 2d d 2 (10) ′ ′ ′ ′ In (10), the terms containing ui (ui ui ) can be ignored. The above discussion and equations apply to the most part of the fluid(gas) field in the canister. However, the gas flow near the wall of the canister may be a little different from the others due to the elastic moving particles. The particles in the gas flow will bounce off the canister wall when they hit the bottom or the sidewall. Because the jetflow will hit the wall several times, we have to study the extra influence of the particles on the gas flow near the wall of the canister.
(
)
[(
(
)
)]
c
b
a (a)
(b)
Due to the inviscid and elastic property, the particle will “deviate” from the “track” along the boundary, which results in the extra force exerted on the gas. Fig.2 describes the motion of particles near the wall. We still use the equation (3) to describe the influence of this extra force Fei′ on the gas. 1 ρu n′ u n′ 2
(11)
3ϕ 3ϕ Fei′ 1 1 1 = C D p ρ u n′ u n′ ≈ C D p ρ • • ui′ui′ mcube 2d p 2 2d p 3 2
[(
)
(
(12)
Because the elastic collision, the ui − u pi term in
)
)]
⎞ ∂k ⎤ ⎟⎟ ⎥ + Gk − ρε ⎠ ∂x j ⎦⎥ (15) ⎡ ⎤ μ ⎞ ∂ε ∂( ρε ) ∂ ( ρεui ) ∂ ⎛ = + ⎢⎜⎜ μ + t ⎟⎟ ⎥ σ ε ⎠ ∂x j ⎥⎦ ∂t ∂xi ∂x j ⎢⎣⎝ (16)
∂ ∂ ( ρk ) ∂ ( ρkui ) + = ∂x j ∂t ∂xi
⎡⎛ μ ⎢⎜⎜ μ + t σ k ⎣⎢⎝
ε2
k + νε where σ k = 1.0 , σ ε = 1.2 , C2 = 1.9 ,
⎛ η ⎞ ⎟, C1 = max⎜⎜ 0.43, η + 5 ⎟⎠ ⎝
η = (2 Eij • Eij )
1
2
k
ε
, Eij =
1 ⎛⎜ ∂ui ∂u j + 2 ⎜⎝ ∂x j ∂xi
⎞ ⎟. ⎟ ⎠
In (16), μt is given by
k2
ε
In (17), the parameter Cμ can be defined as
Cμ =
resisting force equation will be enhanced. Add this term to the motion and energy equations of the gas near the wall.
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(
μt = ρCμ
Take the same steps as before to deal with equation (11).
ρ
(13) The corresponding energy equation is: ∂ ⎡ρ ∂ ⎡ρ ∂p ⎤ ⎤ ui ui + ui′ui′ ⎥ + u j ui ui + ui′ui′ ⎥ + u j + ⎢ ⎢ ∂t ⎣ 2 ∂x j ⎣ 2 ∂x j ⎦ ⎦ ρ ∂ ∂ ⎡ ⎛⎜ ∂ui ∂u j ⎞⎟⎤ ⎛ ⎞ u ′j ⎜ p′ + ui′ui′ ⎟ = μ + ⎢ui ⎥+ 2 ∂x j ⎝ ∂x j ⎢⎣ ⎜⎝ ∂x j ∂xi ⎟⎠⎥⎦ ⎠ ∂ ⎛⎜ ∂ui′ ∂u ′j ⎞⎟ ∂ − ui − ρ ui′u ′j + μ ui′ + ∂x j ⎜⎝ ∂x j ∂xi ⎟⎠ ∂x j ⎛ ∂u ∂u ⎞ ∂u ∂u ′ ⎛ ∂u ′ ∂u ′ ⎞ μ ⎜⎜ i + j ⎟⎟ i − μ i ⎜⎜ i + j ⎟⎟ + ∂x j ⎝ ∂x j ∂xi ⎠ ⎝ ∂x j ∂xi ⎠ ∂x j 3 3ϕ ϕ 4 1 4 1 C D p ρui ui′ui′ + C D p ρ ui′(ui′ui′)′ + 3 2d p 2 3 2d p 2 3ϕ d 3ϕ 1 1 C D′ ρui ui′ui′ + C D′ d ρ ui′(ui′ui′)′ 2d d 2 2d d 2 (14) It is acceptable to choose Realizable k −ε turbulence model [10-17] in calculation in FLUENT. The Realizable k − ε equations are as follows:
+ ρC1 Eij ε − ρC2
Figure 2. Particle motion near the wall
Fei′ = C D Ap
The motion equation near the wall is: ⎞ ∂u ∂u ∂p ∂ ⎛⎜ ∂ui + − ρ ui′u ′j ⎟ + ρ i + ρu j i = − μ ⎟ ⎜ ∂t ∂x j ∂xi ∂x j ⎝ ∂x j ⎠ 3ϕ p 1 3ϕ d 1 4 CD ρ ui′ui′ + C D′ ρ ui′ui′ 3 2d p 2 2d d 2
where
1 A0 + ASU *k / ε
(17)
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A0 = 4.0 ⎫ ⎪ AS = 6 cos φ ⎪ 1 φ = cos −1 ( 6W ) ⎪ ⎪ 3 ⎪ Eij E jk Ekj ⎪ W= ( Eij Eij )1 2 ⎪ ⎬ 1 ⎛⎜ ∂ui ∂u j ⎞⎟ ⎪ Eij = + 2 ⎜⎝ ∂x j ∂xi ⎟⎠ ⎪ ~ ~ ⎪ U * = Eij Eij + Ω ij Ω ij ⎪ ⎪ ~ Ω ij = Ω ij − 2ε ijk ωk ⎪ Ω ij = Ωij − ε ijk ωk ⎪⎭ Parameter Ω is a time mean speed tensor relative to the selected reference frame whose angular velocity is ωk . Realizable k − ε model used to describe the fully developed turbulence is applicable to most part of the flow field in the canister. We have to use different model to solve the flow field near the wall of the canister because the low Reynolds number, turbulence which is not fully developed and viscosity dominate the near-wall region [18]. We choose the low Reynolds number k − ε model proposed by Jones and Launder [19-22] to describe the mentioned region. μ ⎞ ∂k ⎤ ∂ ( ρk ) ∂ ( ρkui ) ∂ ⎡⎛ + = ⎥ ⎢⎜⎜ μ + t ⎟⎟ σ k ⎠ ∂x j ⎥⎦ ∂t ∂xi ∂x j ⎢⎣⎝ (18) 2 ⎛ ∂k 1 2 ⎞ ⎟ + Gk − ρε − 2 μ ⎜⎜ ⎟ ⎝ ∂n ⎠
∂ ( ρε ) ∂ ( ρεui ) ∂ + = ∂t ∂xi ∂x j
⎡⎛ μ ⎢⎜⎜ μ + t σε ⎣⎢⎝
ε
k2
ε
, Cμ = 0.09 , σ k = 1.0
⎛ ∂u ∂u j ⎞ ∂ui ⎟ , f1 = 1.0 , Gk = μt ⎜ i + ⎜ ∂x j ∂xi ⎟ ∂x j ⎠ ⎝ f 2 = 1.0 − 0.3 exp(− Re t2 ) , f μ = exp(−2.5 /(1 + Re t / 50)) , and
Re t = ρk 2 /(ηε ) . The equations we finally need in the present case when we deal with the gas flow in most part of the canister are
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B. Models The physical model of the system can be simplified as follows:
⎞ ∂ε ⎤ ⎟⎟ ⎥ ⎠ ∂x j ⎦⎥
(19) 2 2 μμt ⎛ ∂ 2u ⎞ C1ε ε ε2 ⎜ ⎟ + Gk f1 − C2ε ρ f2 + ρ ⎜⎝ ∂n 2 ⎟⎠ k k where n represents the normal coordinate of the wall, u is the speed parallel to the canister wall, k2 μt = Cμ f μ ρ , C1ε = 1.44 , C2ε = 1.92 ,
σ ε = 1.3 , μt = ρCμ
⎧ ∂ui ⎪ ∂x = 0 ⎪ i ⎪ (9) ⎨ (10) ⎪ ⎪ (15) ⎪ ⎩ (16) The equations near the wall are ⎧ ∂ui ⎪ ∂x = 0 ⎪ i ⎪ (13) ⎨ (14) ⎪ ⎪ (18) ⎪ ⎩ (19)
Figure 3. Meshed physical model
In Fig. 3, the top surface of the model represents the bottom of the missile which is expected to be raised up by the pressure of the gas in the reservoir. The missile will not move until the pressure in the reservoir reaches the critical value. The time this procedure takes varies depending on the inlet flow rate or the inlet pressure. Then the missile moves upwards driven by the gas which is forced into the reservoir continuously from the inlet.
C. Calculations and Conclusions The values of the inlet pressure which are derived from test data, the parameters of the moving zone(dynamic mesh) [23] which is described by the DEFINE_CG_MOTION() macro and the source terms we added to the governing equations are defined by the UDFs(User Defined Functions) [24,25].Below is an excerpt from the moving zone UDF. DEFINE_CG_MOTION(piston,dt,vel,omega,time,dti me) { Thread *t; face_t f;
JOURNAL OF COMPUTERS, VOL. 8, NO. 5, MAY 2013
Message("time=%f,z_vel=%f,force=%f\n",time,v_prev ,force); vel[2]=v_prev; } ... The flow field conditions of the x=0 section are shown in following figures. Fig. 4 and Fig. 5 show the total pressure at 0.07s and 0.36s respectively in the reservoir, while the velocity field under the missile at 0.36s is given in Fig. 6.
makes the pressure drop a little. With the continuous gassteam injection, the stable and enough pressure near the missile is maintained until the missile is pushed out of the launcher.
Figure 6. Velocity field at 0.36s
The gas is injected into the reservoir at a very high speed. Then it rushes towards the bottom of the missile with several eddies in the middle. When it approaches the bottom, its field of velocity becomes uniform. veloctiy
calculations test data
40 velocity (m/s)
real NV_VEC(A); real force,dv; NV_S(vel,=,0.0); NV_S(omega,=,0.0); if (!Data_Valid_P()) return; t=DT_THREAD(dt); force=0.0; begin_f_loop(f,t) { F_AREA(A,f,t); force+=F_P(f,t)*NV_MAG(A); } end_f_loop(f,t) ....
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30 20 10 0 0
0.2
0.4 time (s)
0.6
0.8
Figure 7. Velocity distribution curves
Figure 4. Pressure field at 0.07s
Fig. 7 shows the velocity distribution curves of the missile. According to the calculations(the blue curve), it takes 0.10s for the pressure in the reservoir to exceed the sum of the gravity of the missile and the resisting force which are calculated in the corresponding UDF. After 0.25 seconds, the velocity curve becomes almost linear, which indicates the uniform acceleration and shows good agreement with the pressure variation near the missile bottom. When the missile departs from the canister at 0.65s, its velocity is 37.2m/s. This curve is in conformity with the test data acquired from the experiments(the pink). calculations test data
Figure 5. Pressure field at 0.36s
The pressure near the inlet tops the whole pressure field in the reservoir. At an early stage of the ejection, the pressure in the reservoir increases rapidly because of the small reservoir volume and the missile at rest. Then the gas in the reservoir expands along with the moving missile, which © 2013 ACADEMY PUBLISHER
acceleration (m/s2)
acceleration 120 100 80 60 40 20 0 0
0.2
0.4 time (s)
0.6
Figure 8. Acceleration distribution curves
0.8
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JOURNAL OF COMPUTERS, VOL. 8, NO. 5, MAY 2013
Fig. 8 shows the acceleration-time histories of the missile. The calculations(the blue curve) indicate that the peak value of the acceleration(95.5m/s2) occurs at 0.31s.Then it decreases a little. The final value is 62.8m/s2 with time at 0.63s.These are very close to the values obtained from the experiments(the pink). Both the peak value of the calculations and the max. test data are acceptable according to the material of the missile bottom. III. CONCLUDING REMARKS A simulation has been developed for analyzing the gas dynamics of canister launched missiles. Unique to the technique is the acceleration or the pressure of the missile, which is uniform during the most ejection time. Inclusion of these measures, such as dynamic mesh, k − ε model, and so on, allows a reliable simulation of the ejection. Results of the simulation show that it is in good agreement with test data. This model can be used to analyze the similar launch procedures. REFERENCES [1] Shouzhen Rui and Yuming Xing, “Comparative studies of interior ballistic performance among several missile eject power systems,” Journal of Beijing University of Aeronautics and Astronautics, Vol.35, pp.766–770, June 2009. [2] C. T. Edquist and G. L. Romine, “Canister Gas Dynamics of Gas Generator Launched Missiles,” AIAA-80-1186. [3] C. T. Edquist, “Prediction of the Launch Pulse for Gas Generator Launched Missiles,” AIAA-88-3290. [4] Yu. N. Sadkov, “Numerical simulation of supersonic axisymmetric gas jets,” Computational Mathematics and Modeling, vol. 15, pp. 344–349, April 2004. [5] Vincent Lijo, Heuy Dong Kim, Toshiaki Setoguchi, and Shigeru Matsuo, “Numerical simulation of transient flows in a rocket propulsion nozzle,” International Journal of Heat and Fluid Flow, vol. 31, pp. 409–417, June 2010. [6] D. Albagli and Y. Levy, “Experimental study on confined two-phase jets,” J. Thermophysics, vol. 5, pp. 387–393, July 1991. [7] M. Sommerfeld, “The structure of particle-laden, underexpanded free jets,” Shock Waves, vol. 3, pp. 299– 311, March 1994. [8] Qiang Qi, Qinggui Chen, Yuan Zhou, Haiyang Wang, and Hongmei Zhou, “Submarine-Launched Cruise Missile Ejecting Launch Simulation and Research,” 2011 International Conference on Electronic & Mechanical Engineering and Information Technology, pp.4542–4545, August 2011. [9] Yanhui Tang, Yaoguo Xing, and Chunlong Zhang, “Working Process Simulation of Gas and Steam Launching System,” Journal of Naval Aeronautical and Astronautical University, Vol.24, pp.431–434, July 2009. [10] Xiaochao Zhai, Yajun Chen, and Yi Jiang, “Using dynamic grid to simulate interior trajectory of the launching container,” Modern Defence Technology, Vol.34, pp.24–28, April 2006. [11] B. E. Launder and D. B. Spalding, “The numerical computation of turbulent flows,” Computer Methods in Applied Mechanics and Engineering, pp.269-289, March 1974.
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[12] Shiwu Fang, Xueguang Zhang, and Haiying Qian, “Research on the Valid Powder Energy Coefficient for Canister Type Missile Ejection Launcher,” Missiles and Space Vehicles, pp. 18–24, March 2004. [13] Yi Jiang, Jiguang Hao, and Debin Fu, “3D Unsteady Numerical Simulation of Missile Launching,” ACTA ARMAMENTARII., vol. 29, pp. 911–915, August 2008. [14] B. Raverdy, I. Mary, P. Sagaut, and N. Liamis, “Highresolution large-eddy simulation of flow around lowpressure turbine blade,” AIAA Journal, pp.390–397, March 2003. [15] I. Demirdzic and M. Peric, “Finite volume methods for prediction of fluid flow in arbitrarily shaped domains with moving boundaries,” Int. J. Num. Meth. Fluids, pp.771– 790, October 1990. [16] M. Ishii and K. Mishima, “Two-fluid model and hydrodynamic constitutive relation,” Nuclear Engineering and Design, Vol.82, pp.107–126, March 1984. [17] Hassan, “Mesh generation and adaptivity for the solution of compressible viscous high speed flow,” Int. J. Numerical Methods Engineering, Vol.38, pp.1123–1148, Dec. 1995. [18] R. M. C. So, Y. G. Lai, and H. S. Zhang, “Second-order near-wall turbulence closures: a review,” AIAA Journal, Vol.29, pp.1819–1835, April 1991. [19] C. G. Speziale, “Analytical methods for the development of Reynolds-stress closures in turbulence,” Ann. Rev. Fluid Mech., Vol.23, pp.107–157, March 1991. [20] B. E. Launder, “Second-moment closure and its use in modeling turbulent industrial flows,” International Journal for Numerical Methods in Fluids, Vol. 9, pp.963–985, Sep.1989. [21] B. E. Launder, “Second-moment closure: present and future,” J. Heat Fluid Flow, Vol.10, pp.282–300, April 1989. [22] W. P. Jones and B. E. Launder, “The calculation of lowReynolds-number phenomena with a two-equation model of turbulence,” Int. J. Heat Mass Transfer, Vol.16, pp.1119–1130, June 1983. [23] E. J. Probert, O. Hassan, K. Morgan, and J. Peraire, “An adaptive finite element method for transient compressible flows with moving boundaries,” Int. J. Numerical Methods Engineering, Vol.32, pp.1145–1159, Nov.1991. [24] Fluent Inc., FLUENT User’s Guide, Fluent Inc..USA, 2003. [25] K. F. C. Yiu, D.M.Greaves, S. Cruz, A. Saalehi, and A. G. L. Borthwick, “Quadtree grid generation: Information handling, boundary fitting and CFD applications,” Comput. Fluids, Vol.25, pp.759–769, July 1996.
Yongquan Liu Date of birth: 08.20.1980, male, PH.D in University of Science and Technology Beijing, Research Fields: fluid mechanics, mechanical engineering.
