RESEARCH ON THE BASE CAVITY OF A SUB-LAUNCHED PROJECTILE*

CAO Jia-yi

Department of Mechanics, Shanghai Jiao Tong University, Shanghai 200240, China, E-mail: jycao@sjtu.edu.cn LU Chuan-jing

MOE Key Laboratory of Hydrodynamics, Shanghai Jiao Tong University, Shanghai 200240, China

CHEN Ying, CHEN Xin, LI Jie

Department of Mechanics, Shanghai Jiao Tong University, Shanghai 200240, China

RESEARCH ON THE BASE CAVITY OF A SUB-LAUNCHED PROJECTILE*

CAO Jia-yi

Department of Mechanics, Shanghai Jiao Tong University, Shanghai 200240, China, E-mail: jycao@sjtu.edu.cn LU Chuan-jing

MOE Key Laboratory of Hydrodynamics, Shanghai Jiao Tong University, Shanghai 200240, China

CHEN Ying, CHEN Xin, LI Jie

Department of Mechanics, Shanghai Jiao Tong University, Shanghai 200240, China

The finite volume method based on a multiphase model is adopted to solve the Reynolds-Averaged Navier-Stokes (RANS) equations, which takes into account the effects of fluid compressibility, viscosity, gravity, medium mixture and energy transfer of water and combustion gas. The program Fluent User Define Function (UDF) module combined with the dynamic mesh method is employed to simulate the coupling flow field of combustion gas, water field and trajectory of projectile. The results show that the volume of gas cavity at the bottom of projectile and tail pressure will fluctuate after bottom of the projectile leaving the launch tube. The cause of the fluctuation is analyzed and its effects on the trajectory of projectile are presented. The numerical and experimental results agree well with each other.

sub-launched projectile, gas cavity, trajectory

Introduction

Projectile launch on a submarine is an advanced military technology. Such launching methods possess the advantages of flexibility and elusion. The projectile is accelerated by the high pressure propellant gas and departs from the launch tube. After the projectile bottom passing the deck level, the high pressure gas in tube will pour out and forms a gas cavity between bottom and launch tube which stretches out as the projectile moves upwards. The single gas cavity finally breaks into two parts because of the pressure difference between the gas cavity and ambient pressure. One part remains attached to the tube and the other part goes with the bottom till the projectile out of the water entirely. The latter part interacts with the ambient water to exert an influence on the projectile load and trajectory. Hence, it is significant to study the evolution of gas cavity at the bottom of sub-launched projectile and fluid field inside cavity.

As for the gas cavity at the bottom of an underwater projectile, some related researches on experiments and numerical simulations have been conducted by domestic researchers. Tao et al. (1988) created an experimental system to survey pressure around the tube, and gained the fluctuation pressure data. Lu et al. (1992) adopted an isobaric bubble model to simplify the jet flow ejected into water through a nozzle. The evolution of the combustion bubble was simulated by means of the mixed Eulerian-Lagrangian method and computational technique. Huang et al. (1996) also chose an isobaric bubble model and adopted the Boundary Element Method (BEM) to simulate the collision between gas bubble and skirt-shaped body. Wang et al. (1997) proposed a one-dimensional, unsteady gas flow model combined ideal fluid model to calculate the flows both outside and inside the exhausted gas cavity which is formed behind an underwaterlaunched missile. Shan et al.[1], Cheng and Liu[2]and Wang and Zhao[3]researched on the relation between the angle of a nozzle and the moment of a missile. Zhang et al.[4]investigated the characteristics of water flow field and the pressure variation on weapon surface in process of weapon underwater launching. Li and Lu[5]applied Rayleigh-Plesset equation to establish the model of combustion gas bubble and investigated the fluctuation of the pressure inside bubble. Wang et al.[6]simulated the ejection process of a submarine-launched missile and analyzed the pressure pulation near the canister outlet. He et al.[7]simulated the evolution of the natural caivty and combustion gas of an underwater vehicle and analyzed the interactions between them.

The articles mentioned above are almost focused on heat emission which means the missile carries engines and is impelled by them. The research about cold emission that missile is driven by the projectile gas inside the launching tube has seldom been reported in recent literatures.

The computational models mentioned above were all simplified to some extent. In this article, a more complete computational model is established to simulate the cold emission process, which takes into account the effects of fluid compressibility, viscosity, gravity, medium mixture and energy transfer of water and combustion gas. The coupled flow field of the gas bubble, external water field and the motion of a projectile is simulated to analyze the evolution of the gas bubble at the base of a projectile and its effects on trajectory.

1. Mathematical model

1.1 Governing equations

The process of an underwater projectile launch is complex, which involves unsteady multiphase flow and the coupling calculation between flow field and projectile motion. The high speed of the projectile causes pressure decrease around the shoulder. When the pressure there reduces to be lower than the saturated vapor pressure, cavitation will occur in water. In addition, launch tube is filled with the non-condensable gas before departure, part of which will adhere to low pressure area around the shoulder of the projectile and form an attached gas cavity. Therefore, the mixture model[8,9]coupling with the cavitation model is adopted to simulate such a complex multiphase flow which includes non-condensable gas, vapor and liquid. For this transition multiphase turbulent flows, the equations of continuity, momentum, energy, state and volume fraction in the fluid region are as follows:

Continuity equation

ρ is the density of the mixture, which is defined as

where αkis the volume fraction of phase k andn represents the number of phases. There holds

Moment equation

where Fiis a body force, and μ is the viscosity of the mixture

Energy equation

where g, v and l represent gas, vapor and liquid respectively, and keffis the effective conductivity (k+kt), in which ktis the turbulent thermal conductivity, defined according to the turbulence model being used. The first term on the right-hand side of Eq.(5) represents energy transfer due to conduction andSEincludes any other volumetric heat sources.

State equation

1.2 Cavitation modeling

In cavitating flows, the density of the mixture medium varies across thecavitation influenced region, thus the above described governing equations are not self-contained with the absence of density equation. Therefore, the mathematical model should be complemented with additional cavitation model to take density variation into account. The Zwart-Gerber-Belamri model[10]is taken as the cavitation model here, and it can be expressed as

where the termsReeandRcaccount for the mass transfer between the vapor and liquid phases in cavitation.

where Pvis the saturated vapor pres sure,rBis bubble radius,αnuc.the nucleation site volumefraction, Fvap.the evaporation coefficient, and Fcond.is the condensation coeffient.

1.3 Turbulence modeling

In[11,1t2h]e current research, the RNGk-ε model is adoptedto simulate the turbulence effect.

Turbulent kinetic equation

Dissipation rate equation

where Gkand Gbrepresent the generation of turbulence kinetic energy due to the mean velocity gradientsand buoyancy respectively.YMrepresents the contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate. C1ε, C2εand C3εare constants. σkand σεare the turbulent Prandtl numbers for k and ε, respectively. The turbulent viscosity, μt, is computed by combiningk and ε as follows

1.4 Motion equation of a projectile

During the process of a projectile launch, the loads ontheprojectile body include gravity G, acting forces Fgand Flproduced bygasand liquid around respectively[13], which can be obtained from integrating pressure on the discretized faces of the projectile surfa ce. Thus, the motion equation of a projectile can beexpressed as

where mis themass of the projecti le. The Fluen t User Define Function(UDF) is used to calculate the loads on the projectile bodyand update the velocity of the projectile.

2. Numerical methods

The CFD commercial solv er Fluent is used in th is simulation. The finite vol ume method is ad opted to discretize the governi ng equ ations. Th e cou p ling of the velocity and pressure terms in momentum equationis resolved by the well-known SIMPLE algorithm[14].The second-order upwind discretization scheme is chosen for momentum, energy, turbulent kinetic energy and dissipation energy and the Quick scheme[15]is introduced to discretize the volume fraction.

Fig.1 Sch ematic of a projectile launch

Themoving me sh method is chosen to update the computationalfieldwhichvarieseverytimestepdueto the change of relative position between the projectile and tube.

Fig. 2 Evolution of cavities during lauching process

Fig.3 Pressure evolution during lauching process

Boundary conditions

The whole process ofa vertical launch of a sublaunched projectile by means of a compressed gas is simulated. The process of launch can be divided into several successive stages. Here, the stage after projectile bottom leaving the tube is the one which we pay close attention to. In this stage, a gas cavity is formed between the missile bottom and tube, and finally breaks into two parts. More details focused on the gas cavity attached to the projectile bottom will be presented. According to the computational model of the projectile body and boundary conditions are axisymmetric, the axisymmetric model is chosen as show in Fig.1.

The surfaces of the projectile, tube and deck are set as wall, the Boundary Conditions (BCs) there are as follows

where the subscriptsf and wrepresent fluid and wall respectively.

The bottom of the tube isthe inlet of high pressure compressed gas, which generates the driving force to push a projectile out of the tube. The BCs thereare set as

where p(t) is the pressure obtained from experimental data. The UDF is used to implement the pressure variations withtime. The BCs of the water field is set as

where ρlis the density of water, g the acceleration o fgravity, and h the depth of local.

Figure 2 shows the launch process after projectile bottom leaving the tube, where t0is the characteristic time. In Fig.2, it shows the evolution of the cavity around projectile shoulder which comprises vapor and non-condensable gas andthe gas cavity behind bottom. When the bottom passes through the deck level, the high pressure gas in tube pours out because of the pressure difference between gas and water around. The projectile bottom and the launch tube are connected by a single gas cavity which stretches out axially and expands radially as the projectile moves upwards. With the volume of gas cavity getting larger, the pressure inside cavity decreases. Due to inertial effects it keeps expanding even for the gas pressure lower than

3. Results and analyses

the hydrostatic pressure. It can be observed in Fig.3, whereoP is the characteristic pressure. When the gas cavity expands to a certain extent and the gas pressure is lower than the ambient pressure, it begins to shrink and breaks into two parts finally. The fast radial flow near the breaking location induces a high pressure region and brings two strong transient re-entering jets upwards and downwards respectively. The upward one moves towards and finally impacts the projectile bottom to form a high pressure point at the bottom center. The pressure of bottom center is compared to the average pressure on the bottom in Fig.4. A pressure peak appears when the strike occurs. At the same time, the downward one happens in the part of the cavity that remains attached to the tube. Meanwhile water around the tube squeezes the gas cavity and pours into the launch tube.

Fig.4 Pressure at bottom center compared to average pressure on bottom

Fig. 5 Evolution of the repetition of expanse and shrink of gas cavity

Fig.6 Average pressure on missile bottom

Along with the cavities shrinking, gas pressure inside the cavity increases. Also because of inertial effects gas cavities continues to shrink when the gas pressure higher than the ambient pressure. And such a positive pressure difference causes another repetition of expanse and shrink as shown inFig.5. During the launch process after projectile bottom leaving the tube, the expanses and shrinks of a trailing gas cavity correspond to gas pressure decreases and increases respectively. The curves in Fig.6 present such characteristics.

Fig.7 Streamlines distribution around the bottom

Figure 7 shows the streamlines distributions around bottom on 17/40t0and 21/40t0. In Fig.7(a), are-entering jet is hitting the bottom, and forming a high pressure region in the centerof bottom. Although the average pressure of bottom is not veryhigh at that time, the pressure at the center is as almost four times as the average pressure (see Fig.4). So, more attention should be paid to avoidinginstruments damage in this area. In Figs.7(a) and 7(b), the motion states of cavity surfaces appear differently. The cavity surface is moving inward axially in 17/40t0and outward axially in 21/40t0. Which correspond to the state of a shrinking and an expanding respectively.

In Fig.6 the solid line represents experimental data of average pressure on the missile bottom, and dashdotted line represents numerical results. The pressure on the projectile bottom fluctuates and the fluctuationamplitude of pressure decays during launch process because of energy consumption. The characteristics of numerical result are consistent with the experimental results. And it also can be seen the numerical data compares well with experimental data.Furthermore, the fluctuation of pressure on bottom leads to the fluctuation of gross load on projectile which will have an effect on trajectory.

4. Conclusions

The vertical launch of a sub-launched projectile is complicated which involves phase transition, interations between different phases andthe coupling between flow field and trajectory. For this problem, a numerical computation method based on the multiphase model has been constructed to simulate the launch process. More details on trailing gas cavity have been approached through numerical simulation to analyze its evolution and effects on a projectile trajectory. And the whole evolutive process of a gas cavity can be divided into the following stages.

In the first stage, high pressure gas pours out of the tube and forms a single gas cavity conneting missile and tube after the projectile bottom leaving the tube. Then, the gas cavity expands because of the pressure difference between gas and ambient water, which causes a pressure decrease inside cavity.

In the second stage, as was mentioned above, the expansion delay induces pressure inside cavity lower than ambient one. The negetive pressure difference drives gas cavity to shrink. And finally a single gas cavity is split into two distint cavities. The upper one forms a trailing cavity. The lower one retracts towards the launch tube. Simultaneously, two re-entering jets are found in sperate cavities. The upward one finally strikes the missile bottom and forms a high pressure region.

During the following third phase, the phenomina of cavity expanse and shrink repeats again and again until the projectile leaves water entirely. This characteristic induces the fluctuation of load on projectile bottom and brings an influence to the trajectory.

Inthe present article, the numercial results are compared with the experimental data, which are in good agreement with experimental results, thus providing the important reference for trajectory predition.

[1]SHAN Xue-xiong, YANG Rong-guo and YE Qu-yuan. Fluid forces on a missile with control system of vectorial thrust[J]. Journal of Shanghai Jiaotong University, 2001, 35(4): 625 -629(in Chinese).

[2]CHENG Yong-sheng, LIU Hua. The effect s of nozzle deflection angles on a missile launched underwater[J]. Journal of Hydrodynamics, Ser. A, 2007, 22(1): 83- 92(in Chinese).

[3]WANG Jian-ru, ZHAO Shi-chang. Computations for solid rocket moter tail flow underwater[J]. Journal of Solid Rocket Technology, 2007, 30(5): 388-391(in Chinese).

[4]ZHANG Hua-dong, ZHAO Guo-hua and ZHANG Xiaofang, Simulation of weapon underwater launching with computational fluid dyanmics software[J]. Torpedo Technology, 2009, 17(5): 67-71(in Chinese).

[5]LI Jie, LU Chuan-jing. The model of combustion gas bubble of submarine-launched missile and numerical simulation[J]. Journal of Ballistics, 2009, 21(4): 6-8(in Chinese).

[6]WANG Han-ping, YU Wen-hui and WEI Jian-feng. simulation of pressure field near canister outlet for underwater-launched emulated missile[J]. Acta Armamentarii, 2009, 30(8): 1009-1013.

[7]HE Xiao, LU chuan-jin and CHEN Xin. The scale effect of natural cavitating flow and exhausted gas of an underwatervehicle[J]. Chinese Journal of Hydrodynamics, 2010, 25(3): 367-373(in Chinese).

[8]CHEN Xin, LU Chuan-jing and LI Jie et al. The wall effect on ventilated cavitating flows in closed cavitation tunnels[J]. Journal of Hydrodynamics, 2008, 20(5): 561-566.

[9]ZHANG Ling-xin, ZHAO Wei-guo and SHAO Xueming. A pressure-based algorithm for cavitating flow computations[J]. Journal of Hydrodynamics, 2011, 23(1): 42-47.

[10]ZWART P. J., GERBER A. G. and BELAMRI T. A two-phase flow model for predicting cavitation dynamics[C].Fifth International Conference on Multiphase Flow. Yokohama, Japan, 2004.

[11]ZHOU Ling-jiu, WANG Zheng-wei. Numerical simulation of cavitation around a hydrofoil and evaluation of an RNG k-εmodel[J]. Jounral of Fluids Engineering, Transactions of the ASME, 2008, 130(1): 0113021-0113027.

[12]ZHANG Bo, WANG Guo-yu and ZHANG Shu-li et al. Evaluation of a modified RNG k-ε model for computations of cloud cavitating flows[J]. Transaction of Beijing Institute of Technology, 2008, 28(12): 1065- 1069(in Chinese).

[13]CAO Jia-yi, LU Chuan-jin and CHEN Xin et al. Flow behavior during uncorking process of an underwater hot-launched missile[J]. Journal of Solid Rocket Technology, 2011, 34(3): 281-284, 294(in Chinese).

[14]ASLAM BHUTTA N. M., HAYAT N. and BASHIR M. H. et al. CFD application in various heat exchangers design: A review[J]. Applied Thermal Engineering, 2012, 32: 1-12.

[15]KREJCI V., KOSNER J. Large eddy simulation of jetwall interaction, experimental fluid mechanics 2009[C]. International Conference on Experimental Fluid Mechanics. Liberec, Czech, 2009.

October 5, 2011, Revised January 6, 2012)

* Project supported by the National Natural Science Foundation of China (Grant No. 10832007), the Shanghai Leading Academic Discipline Project (Grant No. B206).

Biography: CAO Jia-yi (1981-), Female, Ph. D. Candidate