AJCPP 2008 March 6-8, 2008, Gyeongju, Korea
Supersonic and Subsonic Projectile Overtaking Problems in Muzzle Gun Applications
Rajesh Gopalapillai, Suryakant Nagdewe and Heuy Dong Kim School of Mechanical Engineering, Andong National University, 388, Songchun-dong, Andong, Korea [email protected]
Toshiaki Setoguchi Department of Mechanical Engineering, Saga University, 1, Honjo, Saga 840-8502, Japan
Keywords: Blast Wave, Projectile, Ballistic Range, Overtaking, Chimera Mesh
Abstract
A projectile when passes through a moving shock ambient gas that has early been perturbed by the blast wave, experiences drastic changes in the aerodynamic wave. Behind the contact surface, an unsteady jet is forces as it moves from a high-pressure region to a developed by the discharge of the compressed gas. low pressure region. These sudden changes in the This unsteady jet may often be accompanied by a forces are attributed to the wave structures produced shock wave, the strength of which is again dependent by the projectile-flow field interaction, and are on the projectile acceleration4. responsible for destabilizing the trajectory of the Following the unsteady jet, the projectile is projectile. These flow fields are usually encountered discharged, consequently leading to strong interaction in the vicinity of the launch tube exit of a ballistic between the projectile and the unsteady jet when the range facility, thrusters, retro-rocket firings, silo former passes through the preceding unsteady jet. The injections, missile firing ballistics, etc. In earlier effects of this interaction are usually reflected in the works, projectile was assumed in a steady flow field unsteady fluctuating forces that act on the projectile when the computations start and the blast wave itself5. As the projectile’s motion proceeds, a bow maintains a constant strength. However, in real shock wave may be produced in front of the projectile, situations, the projectile produces transient effects in depending on the projectile speed and local flow the flow field which have a deterministic effect on the states. Therefore, the projectile flow field, being time- overtaking process. In the present work, the dependent, has a strong dependence on the preceding overtaking problem encountered in the near-field of PBW and bow shock wave. As time advances, the muzzle guns is investigated for several projectile projectile may or may not overtake the preceding Mach numbers. Computations have been carried out PBW in a short distance, subject to certain conditions using a chimera mesh scheme. The results show that, which will be discussed later, in more detail. the unsteady wave structures are completely different Meanwhile, the high-pressure gas which, in from that of the steady flow field where the blast wave general, is obtained by the combustion explosion in maintains a constant strength, and the supersonic and the ballistic range6, and used to drive the projectile subsonic overtaking conditions cannot be initially, is discharged, immediately after the distinguished by identifying the projectile bow shock projectile is ejected from the launch tube. The wave only. secondary blast wave (SBW) is formed due to the sudden discharge of the high-pressure combustion gas. Introduction The so called Ritchmyer-Meshcow instability develops when the contact surface following the PBW When a projectile suddenly starts to move in the is swept by the SBW7. launch tube of a ballistic range, a series of A highly under-expanded supersonic jet is compression waves are formed ahead of the projectile produced due to the discharge of the high-pressure due to its piston effect. These compression waves combustion gas. Inside the jet, a strong Mach disk is coalesce into a normal shock wave in which its formed to match the pressure levels behind the strength depends on the acceleration of the projectile. moving projectile. If the discharged high-pressure The shock wave propagates towards the exit of the combustion gas maintains a constant mass flow rate, launch tube, and is discharged from the launch tube, the under-expanded jet structure will be steady8, forming a spherical blast wave due to the shock unlike the unsteady jet produced by the discharge of diffraction at the exit of the launch tube. This blast the compressed gas in between the normal shock wave is usually called the primary blast wave (PBW) wave and the projectile in the very beginning9. and develops with associated starting vortices at the As described above, the flow field around the exit of the launch tube1-3. moving projectile is highly complicated as well as Shortly after the discharge of the shock wave strongly time-dependent. There are various types of from the launch tube, the compressed gas in between interactions among the shear layers, shock waves, the shock wave and moving projectile inside the wake flow behind the projectile and the moving launch tube is discharged and a contact surface is projectile itself. These complex flow phenomena formed at the boundary of the compressed gas and the 711cannot help but be reflected on the aerodynamic 1
AJCPP 2008 March 6-8, 2008, Gyeongju, Korea forces, thus having major influences on the flight with constant strength. The overtaking process being control and stability of projectile10. purely time-dependant, many questions need to be All these related phenomena take place in a very answered with regard to its transient nature. It is not short interval of time, and the measurement of the known exactly at what conditions the projectile detailed aerodynamics is extremely difficult. So far, overtakes the blast wave and what parameters control the investigation of the projectile aerodynamics and the overtaking problem. Moreover, detailed projectile aeroballistics has been limited to the visualization aerodynamics produced at the instant when it pictures of the flow field using a high-speed optical overtakes the blast wave is an exciting problem to the system11, and quantitative data associated with the aerodynamic designers. flying projectile aerodynamics are sparse to date. It is The present study concerns with the projectile expected that the aerodynamic forces drastically aerodynamics and gets insight into the detailed change as the projectile interacts with the unsteady jet overtaking flow field. One-dimensional analysis has and the shock systems around or it passes through the been performed to shed new light on the overtaking primary blast wave12. For example, as the projectile flow fields. A computational fluid dynamics (CFD) passes through the primary blast wave, or the method using a moving grid system has been projectile interacts with the preceding unsteady jet, it employed to simulate the projectile aerodynamics on experiences a sudden change in the aerodynamic drag. such flow fields. The present computations The aerodynamic force data associated with such successfully represent the projectile aerodynamics situations are highly lacking and have been characteristics and predict well the projectile ambiguously conjectured until now. acceleration, velocity and aerodynamic drag which Only a few works9-12 have been to date made to were hardly obtained in experiments. investigate such interaction flow field around the flying projectile. Thus, the prediction of the unsteady One –Dimensional Analysis of the Projectile projectile aerodynamic data is now one of very Overtaking challenging works which are recently being paid In general, the speed of sound downstream of a much attention from researchers and aerodynamic shock wave is higher than that upstream of it. Thus, a designers as well. wave or an object can overtake the preceding shock Of many aspects of the unsteady projectile wave, depending on the strength of the shock wave aerodynamics described above, one of the most and the speed of the object relative to the flow behind important phenomena occurs as projectile overtakes the shock wave. In order to analyze the projectile the preceding blast wave. Recently, Watanabe, et. al.13 overtaking problems, we assume the overtaking flow investigated the projectile overtaking flow fields to to be unsteady and one-dimensional. The primary study the projectile aerodynamics. They used a one- blast wave is hence assumed as a normal shock wave dimensional analysis to study the overtaking criteria with constant propagation speed. For a given normal and argued that the overtaking process can be either shock wave that propagates at constant speed of Mach subsonic or supersonic based on the projectile Mach number Ms in still air, the flow properties can be number relative to the flow behind the blast wave. calculated behind the shock wave. It is also assumed They arbitrarily assumed the projectile and blast wave that a projectile with speed up is propagating towards Mach numbers and their computations started from a the normal shock wave. steady state solution of the flow field. The blast wave Assuming that the flow states across the shock maintained constant strength throughout their wave are denoted by the subscripts 1 and 2, and the 14 computations. Later, Ahmedika and Shirani have projectile velocity is given by up, the shock wave done viscous computations on transonic and supersonic overtaking flow fields using similar assumptions. In both of these works, the relative projectile Mach number was the major factor to decide the overtaking criterion. However, in actual situations where the blast wave propagation is a transient phenomenon, the strength of the blast wave will be decreasing and the relative projectile Mach number will no longer be constant during the whole overtaking process. In such a situation, neither the flow fields nor the projectile aerodynamic characteristics can be determined accurately based on the relative projectile Mach number. Moreover, the transient structures of the flow field will have a deterministic effect on the aerodynamic characteristics of the projectile. Hence, in real situations where the spherical blast wave is propagating, the overtaking process will be Fig.1. Schematic diagram showing the projectile completely different from that of a steady blast wave motion relative to the blast wave 712 2
AJCPP 2008 March 6-8, 2008, Gyeongju, Korea
u u p line indicates the flow state, in which M p1 = M s , i.e, M = s , M = , (1) s p1 the boundary for the projectile to just catch up the a1 a1 moving shock wave. Below this line, overtaking is The projectile Mach number M relative to the flow p2 impossible as the projectile velocity is less than the state behind the moving shock wave is obtained by shock wave velocity. The dotted line ( M p2 =1) u p − u2 M p2 = (2) demarcates the supersonic and subsonic overtaking a2 regimes. In the case of supersonic overtaking, the In Eq.(2), the projectile motion is supersonic relative projectile velocity with respect to the flow behind the to the flow state behind the shock wave, if Mp2 is shock wave is supersonic and hence the wave system larger than 1.0 and is subsonic if Mp2 is less than 1.0, produced by the motion of the projectile cannot travel and Eq.(2) can be written as upstream of it. In this situation, the shock wave will not be affected by the perturbations produced by the a1 M p2 = M p1. − M 2 (3) wave system associated with the projectile until the a2 projectile catches it up. Another important feature of Hence, for the supersonic overtaking, the supersonic overtaking condition is the generation of a bow shock wave just ahead of the projectile. a2 M p1 > ()1+ M 2 (4) If the relative projectile Mach number with a1 respect to the flow behind the shock wave is less than and for subsonic overtaking, 1.0, the wave system associated with the projectile can travel upstream and the shock wave will be a2 disturbed before the projectile reaches up to it. This is M p1 < ()1+ M 2 (5) a1 6
For a moving shock wave, the Mach number M2 and the speed of sound a2 of the flow behind the shock 5 Supersonic Overtaking wave can be expressed as a function of Ms, Mp2 > 1 1 1 ⎛ ⎞ Mp2 < 1 2 = ⎜ ⎟ 2 2 ⎡⎛ 2 γ −1⎞⎛ γ −1 2 ⎞⎤ p (6) M M 2 = ⎜ M s − ⎢⎜γM s − ⎟⎜ M s +1⎟⎥ ⎟ 4 M s 2 2 = ⎜ ⎣⎝ ⎠⎝ ⎠⎦ ⎟ g 1 in M p ⎝ ⎠ ak rt ve p1 O 1 3 c
M i ⎡ 2 ⎤ 2 on a T ⎛ 2γ 2 ⎞⎛ 2 + ()γ −1 M s ⎞ s 2 = 2 = ⎜1+ M −1 ⎟⎜ ⎟ (7) ub ⎢⎜ ()s ⎟⎜ 2 ⎟⎥ S a1 T1 ⎣⎝ γ +1 ⎠⎝ ()γ +1 M s ⎠⎦ Overtaking not possible 2 M < M Thus, Eq.(3) becomes p1 s
M p1 1 M p2 = 1 ⎡⎛ 2γ ⎞⎛ 2 + ()γ −1 M 2 ⎞⎤ 2 ⎜1+ M 2 −1 ⎟⎜ s ⎟ ⎢⎜ ()s ⎟⎜ 2 ⎟⎥ ⎣⎝ γ +1 ⎠⎝ ()γ +1 M s ⎠⎦ 0 123456 ⎛ 1 ⎞ M ⎜ ⎡ γ −1 γ −1 ⎤ 2 ⎟ s 2 ⎛ 2 ⎞⎛ 2 ⎞ (8) − ⎜ M s − ⎢⎜γM s − ⎟⎜ M s +1⎟⎥ ⎟ ⎜ ⎣⎝ 2 ⎠⎝ 2 ⎠⎦ ⎟ ⎝ ⎠ Fig.2. Flow regimes of moving projectile and
When Mp2 is greater than 1.0, the projectile is moving blast wave (Ms: Blast wave Mach number, Mp1: supersonically relative to the flow behind the shock Projectile Mach number and Mp2: Projectile Mach wave (or the blast wave) with its Mach number Ms. number relative to the flow behind the blast wave) Meanwhile, when Mp2 is less than 1.0, the projectile is moving with a subsonic velocity with respect to the referred to as the subsonic overtaking condition, and flow behind the blast wave. The conditions in which no bow shock wave is generated ahead of the Mp2 is greater than 1.0 or less than 1.0 are referred to projectile due to the subsonic motion when it is as supersonic and subsonic overtaking conditions moving behind the preceding shock wave. respectively. If the projectile Mach number is less However, in actual flow fields where the than the blast wave Mach number (Mp1 AJCPP 2008 March 6-8, 2008, Gyeongju, Korea 3 which those cells in a particular zone which are overlapped by a wall boundary of another overset zone are blanked and the chimera-boundary cells are 2.5 Overtaking possible Overtaking impossible identified. In the present computations, a set of multi- Mp =3.0 1 block grids around the projectile (minor grids) is 2 constructed and these surrounding grids that are to 2.5 move typically overlap on the background grids (major grids) of the launch tube and near-field. 2 P 1.5 Subsequently, a communication between the chimera M 2.0 boundary cell-centroids and the overlapping cell- Mp2=1 Supersonic overtaking nodes is established through a tri-linear interpolation. 1 The required parameters are then calculated in all 1.5 Subsonic overtaking cells for each time step. 1.25 In computations, projectile is identified as the 0.5 1.0 moving body and is modeled with a motion of 6- degree of freedom (DoF). The 6-DoF motion requires 0 the Euler’s equations of motion (6 scalar equations) to 1234 be numerically solved at each time step to obtain the M s displacement and velocities from the force and Fig.3. The overtaking flow regimes based on relative moments. The physical information required for projectile Mach number simulating 6-DoF motion in the present problem are, mass of the projectile, initial center of gravity in the number Mp2 with respect to the shock wave Mach inertial frame of reference, mass moment of inertia in number Ms for several values of projectile Mach the axis system of the projectile, orientation of the number Mp1. The shaded area refers to the flow projectile axis system with respect to the inertial regime that the projectile cannot overtake the shock frame of reference, constraints on motion and the wave, and the boundary of two regimes of possible aerodynamic forces and moments acting on the and impossible overtaking conditions represents the projectile which are obtained by solving the domain condition Ms=Mp1 where the projectile just catches up numerically. Based on the 6-DoF, the minor grids the shock wave. The dotted line Mp2=1 demarcates the constructed around the projectile move through the subsonic and supersonic overtaking regimes. background grids on the launch tube and near-field, Here it is again noted that the analytical results communicating the resolved parameters with each described above are obtained based on one- other. dimensional normal shock relationships, where the normal shock wave Mach number and the projectile Computational domain and boundary conditions Mach number remain constant. However, a blast wave Figure 4 illustrates the computational domain, the is, in actual, produced instead of the normal shock boundary conditions and the projectile configuration wave, and its strength attenuates with the propagation used in the present study. The projectile has a length distance. For an initial blast wave with a given of 50 mm and diameter of 20 mm. When the strength Ms, the blast wave reduces with propagation, computations start, a moving shock wave is assumed nearly along a constant Mp1 line in the near field from at the exit of the launch tube where the projectile is the launch tube. This means that the projectile kept inside at a distance of 50 mm behind the shock overtaking criterion cannot be determined using only wave. The flows ahead and behind the projectile are the initial flow conditions. assumed to be in the same condition as that of the In the present study, a computational fluid flow behind the moving shock wave which is at the dynamics method using a moving grid system has exit of the launch tube and the projectile also is been employed to understand the projectile overtaking moving with the velocity of the flow behind the problems which are applicable to actual conditions. A moving shock when the computation starts. The same normal shock wave is assumed at the exit of the flow conditions are assigned to the rear end of the launch tube and has its initial Mach number Ms. In launch tube as inlet boundary conditions. This is a computations, the values of Ms and the projectile quite reasonable assumption as it refers to a condition Mach number Mp1 are varied to investigate the of the compression explosion in a ballistic range projectile overtaking phenomena. facility6. The computations are carried out only up to a point where the projectile is overtaking the blast Chimera mesh wave. During this short duration, the high-pressure The present computations make use of the chamber where the explosion occurs can supply the chimera mesh scheme for structured grids. For flow conditions at a constant rate, and no expansion simulating the projectile motion, this chimera mesh waves from the projectile base reach the rear side of scheme allows the overlapping of one zone over the the launch tube. other. This method uses a hole-cutting algorithm in 714 4 AJCPP 2008 March 6-8, 2008, Gyeongju, Korea Fig.4 Computational domain, projectile configuration and boundary conditions The initial Mach number (Ms) of the shock wave pressure difference. A secondary shock wave is which is at the exit of the launch tube is varied (1.4, forming outside the launch tube, as explained in the 2.0, and 2.5) to get various flow fields. The projectile introduction part, due to the shock wave diffraction at Mach number and the flow conditions behind the the launch tube exit. A sudden drop in acceleration is shock wave have been calculated from one- observed at state B as the projectile is interacting with dimensional normal shock relations for a moving the secondary shock wave. shock wave. 15000 Results and Discussion 10000 Validation of the results CFD by Jiang et al.7 Present Computations In order to examine the numerical accuracy of the ) 2 s present computational method, a particular case from / (Van Leer's Higher Order) m 5000 ( Ref.7 is analyzed. The Mach number (Ms) of the n o i t shock wave at the exit of the launch tube is 5.0 when a r e l the computation starts, and the Mach number of the e 0 c c projectile (M ) is 4.0. Projectile mass is 50 gms. As a p e l i seen from Fig.5, the projectile acceleration history t c e agrees very well with that of Ref.7. Figure 6 shows j -5000 o r the comparison of the flow field of the present P computational method with the numerical schlieren -10000 picture of another case in Ref.7. Here, Ms=3.0, and Mp=2.2. The locations of projectile, blast wave and bow shock are matching very well with the numerical -15000 schlieren pictures. It is also seen that the contact 0 100 200 300 400 discontinuities, vortical structures, supersonic jet, t (μs) Mach disc and the focusing shock wave behind the Fig.5 Comparison of projectile acceleration histories projectile are very well captured using the present computational method in comparison with the During this period, the projectile is accelerating numerical schlieren. slightly from state C to state D when it has passed Figure 7 shows the acceleration history of the through the secondary shock wave. It can be seen that projectile having a Mach number (Mp1) equal to 1.75. after the projectile passed through the secondary The Mach number of the shock wave at the launch shock wave, a bow shock wave is forming in front of tube exit (Ms) is 2.5. When the projectile is inside the the projectile as it travels from a low velocity flow launch tube, projectile velocity (up), “post-shock” behind the secondary shock wave to a high-velocity velocity (u2) and the flow velocity behind the flow behind the PBW. At this period, Mp2 becomes projectile are the same, and hence projectile greater than 1.0 as will be explained later. acceleration is zero. At state A, shown in Figs. 7 and At state D, the projectile is on the verge of 8, the head of the projectile is coming out of the discharge from the launch tube, and experiences a launch tube and it starts accelerating due to the 715 5 AJCPP 2008 March 6-8, 2008, Gyeongju, Korea 6000 B 4000 D C 2000 ) 2 s / m ( p a A 0 G F H -2000 E -4000 0 0.2 0.4 0.6 t (ms) Fig.7 Acceleration history of a cylindrical projectile (Mp1=1.75, Ms=2.5) Fig.6 Comparison of iso-density contours with numerical schlieren One notable difference is the absence of the secondary shock wave in the jet in front of the sudden retardation due to the expansion of the gases projectile. This is due to the weaker diffraction the from the launch tube behind the projectile. From the shock wave at the launch tube exit. Another state E onwards, projectile experiences fluctuations in observable difference is in the shear layers formed the acceleration due to the effects of a developing along the projectile side walls. In Fig.8, the shear underexpanded jet behind it. It can be seen that a layers are found to be more attached to the projectile Mach disk is forming behind the projectile in the walls than in Fig.9. This is due to the strengths of the underexpanded jet. This is to adjust the pressure bow shock waves produced in each case. In first case levels of flows just behind the projectile and that is where the projectile Mach number is 1.75, the discharged from the launch tube. During the initial strength of the bow shock wave is more as the relative period of the devleopment of the underexpanded jet, velocity of the projectile with respect to the flow unsteady vortical structures can be observed at both behind the blast wave is more as compared to the case the ends of the Mach disk. where the projectile Mach number is 1.25. At state F, the bow shock wave in front of the A weaker shock wave cannot exert more pressure projectile catches up the primary blast wave and there on the shear layers to make them attached to the will be a series of expansion waves travelling in projectile wall and hence for a reduced projectile between the front side of the projectile and the bow Mach number and the blast wave strength, the shear shock wave. This is necessary for the bow shock layers can be found more separated from the wave to adjust its strength consistent with the relative projectile wall. velocity of the projectile and the gas flow around it. The Mach disk formed in the under-expanded jet The slight decrease in the deceleration of the behind the projectile in this case (Mp1=1.25) is found projectile from F to G is due to this excursion of the to be smaller than that in the first case where expansion waves between the bow shock and the Mp1=1.75. This can be attributed to the weaker projectile. expansion of the gas behind the projectile. However, The bow shock has reached the PBW at state G the size of the Mach disk is varying as the projectile and attainted a consistent strength now when it moves further away from it. The moving projectile merges with the PBW and triple points are formed at will develop a low-pressure zone just behind it and each side of the projectile due to the bow shock-blast this will lower the strength of the Mach disk and it wave interaction. The projectile slightly decelerates becomes smaller. But, when the projectile is moving due to the increased pressure and density conditions away, the pressure recovers and the Mach disk starts ahead of the projectile where the flow is stagnant. to grow. This process can be seen through the time This completes the overtaking process, as now the frames from t=0.367ms to t=0.9802ms in Fig.9. Also bow shock wave in front of the projectile is equivalent many shock cells are seen at the end of the launch to the one that is formed on a supersonic body moving tube after the under-expanded jet has become steady. through stagnant air. Figure 9 shows the Mach Figure 10 shows the iso-density contours in the contours of the projectile moving at a Mach number flow field of a projectile with a Mach number of 0.57. of 1.25. Some of the flow field features are similar to The shock wave Mach number Ms is 1.4. Here also that of that of the previous one. the secondary shock wave is absent in the jet in front 716 6 AJCPP 2008 March 6-8, 2008, Gyeongju, Korea A B C D E F G H Fig.8 Iso-density contours corresponding to various time instants in Fig.7 (Mp1=1.75, Ms=2.5) of the projectile due to a weaker expansion. The these cases, the projectile starts accelerating only starting vortices can be clearly seen in front of the when it reaches the exit of the launch tube. The projectile. The flow field exhibits the presence of acceleration history of the projectile with a subsonic many vortical structures behind the projectile also. It Mach number is completely different from that of the should be noted here that the flow behind the blast supersonic case. The acceleration history fluctuates in wave is completely subsonic. During the diffraction a random fashion in this case. This is due to the of the shockwave at the exit of the launch tube, presence of a number of vortices forming behind the expansion waves re-enter the launch tube. These projectile as seen from Fig.10. No bow shock wave is expansion waves will make the projectile accelerating forming here and the projectile never overtakes the even when it is inside the launch tube as shown in blast wave as will be described later. In order to Fig.11 where the projectile acceleration histories are investigate the aerodynamic characteristics of the plotted for all the cases. In the other two cases where projectile, the drag coefficient histories of the the flow behind the blast wave is supersonic, the flow projectile are plotted for various projectile Mach will not re-enter the launch tube when the shock wave numbers, in Fig.12. The coefficient of drag of the is diffracting at the end of the launch tube. Hence for 717projectile (Cd) is defined as 7 AJCPP 2008 March 6-8, 2008, Gyeongju, Korea 2D and hence the coefficient of drag will increase in the Cd = 2 (19) negative direction, giving the projectile a thrust force. γp1M p1 Ap It should also be noted that the drag coefficient in The same trends as that of the acceleration history can the negative direction is maximum for Mp1=0.57 be seen for the Cd curves of the three cases of Mp1. It where the projectile has the least Mach number. As is seen that when the projectile Mach number is mentioned early, for the projectile Mach number of decreasing, the peak value of Cd is increasing in the 0.57, the flow is completely subsonic and the negative direction. Here, it is worth noting that the expansion wave will re-enter the launch tube and drag force experienced by the projectile is entirely interact with the projectile. This reduces the drag on due to the wave drag and pressure drag as the present the projectile and hence the drag coefficient peaks up computation uses the inviscid flow assumptions. For in the negative direction when the projectile is inside reduced projectile Mach numbers, the blast wave is the launch tube as seen from Fig.12. also weak because of the chosen conditions. Hence, In order to identify the motions of projectile and for smaller projectile and blast wave Mach numbers, the blast wave and the overtaking points, the x-t the strength of the shock system produced around the diagram is presented in Fig.13. The blast wave path projectile is weak compared to larger projectile and starts from the exit of the launch tube and the blast wave Mach numbers. This will induce less drag projectile starts from a location 50mm behind the force on the projectiles travelling at reduced velocities shock wave which is at the exit of the launch tube. Fig.9 Iso-density contours (Mp1=1.25 and Ms=2.0) 718 8 AJCPP 2008 March 6-8, 2008, Gyeongju, Korea 0.125 ms 0.2375 ms 0.2600 ms Fig.10 Iso-density contours (Mp1= 0.57 and Ms=1.4) The paths for projectile and the blast wave are projectile drag coefficient history in order to converging and coming very close at certain points understand the overtaking effects in the aerodynamic for the cases of Mp1=1.75 and 1.25. These points are characteristics. In Fig.12, the overtaking points are identified as the overtaking points. For Mp1=1.75 and identified, and it is seen that, at the overtaking points, 1.25 the overtaking points are 0.34ms and 0.835ms there are hardly any changes in the drag coefficients. respectively. It is noted that the blast wave path and For the case of Mp1=1.75, the overtaking process starts projectile path never meet at a specific point. This is from a constant drag where the bow shock catches up because; after the projectile overtakes the blast wave, the blast wave to a slightly increased drag when the there will be a single detached shock wave in front of projectile comes out of the blast flow field. The the projectile. Hence the projectile path will be behind overtaking process starts as a transient phenomenon the shock wave path by the shock stand-off distance due to the excursion of the expansion waves in as seen from Fig. 13. The shock stand-off distances in between the projectile front and the bow shock as the cases of Mp1=1.75 and 1.25 are 15mm and 23 mm described early. This ends up in a steady state after respectively. In the case of Mp1=0.57, the projectile the bow shock is merged with the blast wave and the and blast wave paths are diverging and the projectile projectile overtakes it. This leads to the increment in never overtakes the blast wave. Hence the overtaking the drag coefficient of the projectile immediately after point is at infinity for this case, as seen from Fig.13. it overtakes the blast wave. The overtaking points should be identified in the 719 9 AJCPP 2008 March 6-8, 2008, Gyeongju, Korea 6000 condition (Mp1 ) than the blast wave Mach number (M ). After the 2 s s / blast wave is diminished enough (after point t1), the m ( n overtaking becomes possible. Hence in the whole o 2000 i t a motion of the projectile, the overtaking is impossible r e l e for small time duration. This time duration solely c c a depends on the blast wave attenuation for a given e l 0 i t projectile Mach number Mp1. For larger blast wave c e j Mach numbers (M ), the impossible overtaking o s r P condition lasts for more time for the same projectile -2000 Mach number Mp1. 2 Blast wave -4000 Projectile 00.40.81.2 t (ms) 1.6 Fig.11 Acceleration histories of the projectile for Overtaking impossible various Mach numbers 2 Ms=2.0 1.2 Mp1=0.57 Mp1=1.25 Ms=1.4 (ms) Overtaking point Overtaking 0 t (0.835 ms) 0.8 Ms=2.5 -2 Mp1=1.75 Overtaking point d 0.4 (0.340 ms) C -4 0 Mp1=1.75, Ms=2.5 M =1.25, M =2.0 -0.2 0 0.2 0.4 0.6 -6 p1 s x (m) Mp1=0.57, Ms=1.4 Fig.13 x-t diagram of the projectile for various Mach -8 numbers 0 0.4 0.8 1.2 t (ms) Fig.12 Drag histories of the projectile for various 2.5 Mach numbers Overtaking impossible for Mp1=1.75 Overtaking impossible The increment in the drag of the projectile at the 2 for Mp1=1.25 time of overtaking is a strong function of the blast Overtaking impossible wave Mach number as can be seen from the drag for Mp1=0.57 history of M =1.25. Here the changes in the drag Mp =1.75 p1 1.5 1 coefficient are hardly observable due to a weak blast p2 M M = 1.25 p1 wave. This is one of the major characteristics of the , s real flow field, where, by the time the projectile M overtakes the blast wave, the blast wave will have 1 been diminished in its strength, and will have M =1 Mp1= 0.57 negligible effect on the projectile aerodynamics. p2 In order to clarify the effects of the attenuating 0.5 blast wave on the overtaking process, the variations of Ms M the blast wave Mach number Ms and the relative p2 projectile Mach number Mp1 with time are shown in 0 Fig.14 for the computational cases analysed in the 0t1 t2 0.10.20.30.40.5 present work. From Fig. 14, it can be seen that during t (ms) the whole overtaking process, the blast wave Mach Fig.14 Variation of Ms and Mp2 with respect to time number Ms is varying from an impossible overtaking 720 10 AJCPP 2008 March 6-8, 2008, Gyeongju, Korea For the case of Mp1=1.25 also, the overtaking number Mp2, this Mach number cannot be depended condition changes from an impossible one to a upon for studying the projectile aerodynamic possible one. Here the time duration for the characteristics. The projectile passes through transient impossible condition (up to t2) is more as compared to flow structures and thereby experiences variations in the first one. Hence it is impossible to judge based on relative projectile Mach number. These transient flow initial conditions whether the projectile overtakes the structures are responsible mainly for the drag blast wave in real flow situations as the projectile fluctuations on the projectile rather than the always passes through an impossible overtaking overtaking process. The drag coefficient history condition during the initial period of its motion. confirms that the overtaking process makes little However, for Mp1=0.57, the overtaking condition is difference in its aerodynamic characteristics. This is always in the impossible regime and the projectile because in the real flow field, by the time the never overtakes the blast wave. projectile overtakes the blast wave, the blast wave The subsonic and supersonic overtaking regimes becomes weak and will no longer be able to influence of the projectile can be identified from the variation of the projectile aerodynamics characteristics to a great the relative projectile Mach number (Mp2) in Fig.14. extend. However, it should be noted that, for very For both Mp1=1.75 and 1.25, the relative projectile large projectile and blast wave Mach numbers, the Mach number Mp2 is varying from a value less than 1 projectile overtakes the blast wave within a very short in the earlier part of projectile motion to that greater time and is influenced greatly by the overtaking than 1 in the later part where the blast wave has process as the blast wave do not diminish diminished. This makes the distinction of the flow significantly within this time period. fields almost impossible based on Mp2 as figured out by Watanabe, et. al13. They have shown using the one- Conclusions dimensional theory that the flow fields can be distinguished solely based on the relative projectile A computational study has been performed using a Mach number Mp2. Since they assumed constant blast moving grid method to analyse the effects of the wave strength, the relative projectile Mach number projectile overtaking a moving shock wave on the also was constant and subsonic or supersonic projectile aerodynamic characteristics. A one- overtaking conditions could be computationally dimensional analysis has also been carried out to created. However, in the real situations, as the present identify the projectile overtaking criteria. The computations show, the projectile will pass through a analytical results show that the projectile overtaking mixed situation of both subsonic and supersonic flow flow fields can be in a subsonic or supersonic flow regimes during its travel to overtake the blast wave. It regime based on the relative projectile Mach number. will always have a subsonic motion relative to the However, it is found in the present computations that flow behind the blast wave for a very small duration the actual flow fields cannot be distinguished with during its initial periods of motion and it becomes a relative projectile Mach number only, since the blast supersonic motion later when the blast wave has wave strength is diminishing with time and space. It is diminished in its strength. Thus the subsonic also noticed that the projectile always passes through overtaking regime of the projectile in the actual flow a subsonic flow regime relative to the flow behind the problems is a purely transient phenomenon. In such a blast wave, before it overtakes the blast wave. This situation, the projectile cannot have mere subsonic subsonic overtaking process is a transient overtaking condition as illustrated in the previous phenomenon, and the overtaking process of the work13. The overtaking will always be supersonic as projectile will be supersonic eventually. the blast wave is attenuating with time. The aerodynamic characteristics of the projectile Moreover, it has been shown in the previous work are hardly affected by the overtaking process for that, the main characteristic of the supersonic smaller blast wave Mach numbers as the blast wave overtaking (Mp2 > 1) is a bow shock. However, in the will become weak by the time it is overtaken by the actual flow fields, a bow shock will always be projectile. The projectile drag coefficient is affected produced in later part of the projectile motion as the greatly by the unsteady flow structures through which subsonic overtaking condition always changes to the projectile travels in the near-field, than the supersonic overtaking condition. This can be seen overtaking process. However, for larger projectile and through the projectile flow fields depicted in Figs. 8 blast wave Mach numbers, the overtaking effects can and 9. Thus, it is hardly possible to confirm whether be predominant since the projectile overtakes the blast the overtaking is supersonic or subsonic based on wave within a very short time after it is discharged either the relative projectile Mach number Mp2 or the from the launch tube and overtaking process involves presence of the bow shock wave in front of the transient effects of shock interactions. projectile like the one-dimensional studies can do. For the case of Mp1=0.57, Mp2 lies always in the subsonic References regime. This does not have any meaning as in this case, the projectile never overtakes the blast wave. 1) Kim, H. D., and Setoguchi, T.: Study of the Though the instantaneous local flow fields on the Discharge of Weak Shocks from an Open End of a projectile depend on the relative projectile Mach 721 11 AJCPP 2008 March 6-8, 2008, Gyeongju, Korea Duct, Journal of Sound and Vibration, 226, 1999, M: Mach number pp.1011-1028. Mp1: projectile Mach number relative to still air 2) Kim, H. D., Lee, D. H., and Setoguchi, T.: Study of Mp2: projectile Mach number relative to flow behind the Impulse Wave Discharged from the Exit of a the moving shock wave Right-angle Pipe Bend, Journal of Sound and Ms: shock wave Mach number Vibration, 259, 2002, pp.1147-1161. p: pressure, N/m2 3) Bagabir, A., and Drikakis, D.: Numerical T: temperature, K Experiments Using High-Resolution Schemes for t: time, ms Unsteady, Inviscid, Compressible Flows, Computer u: velocity, m/s Methods in Applied Mechanics and Engineering, 193, 2004, pp.4675-4705. Greek Letters 4) Sun, M., and Takayama, K.: The Formation of a Secondary Shock Wave behind a Shock Wave γ: ratio of specific heats Diffracting at a Convex Corner, Shock Waves, 17, ρ: density, kg/m3 1997, pp.287-295. 5) Sun, M., Siato, T., Takayama, K., and Tanno H.: Subscripts: Unsteady Drag on a Sphere by Shock Loading, Shock Waves, 14, 2005, pp.3-9. p: projectile 6) Rajesh, G., Kim, H. D., Setoguchi, T., and s: shock wave Raghunathan, S.: Performance Analysis and 1, 2: downstream and upstream of moving shock Enhancement of the Ballistic Range, Proc. IMechE, wave Part G: J. Aerospace Engineering, 221(5), 2007, pp.649-659. 7) Jiang, Z., Takayama, K., and Skews, B. W.: Numerical Study on Blast Flow Fields Induced by Supersonic projectiles Discharged from Shock Tubes, Physics of Fluids, 10(1), 1998, pp.277-288. 8) Shapiro, A. H.: The Dynamics and Thermodynamics of Compressible Fluid Flow - Vol.1, The Ronald Press Company, New York, 1953, p.383- 384. 9) Rajesh, G., Kim, H. D., Matsuo, S., and Setoguchi, T.: A Study of the Unsteady Projectile Aerodynamics Using a Moving Coordinate Method, Proc. IMechE, Part G: J. Aerospace Engineering, 221(5), 2007, pp.691-706. 10) Schmidt, E. M., Fansler, K. S., and Shear, D. P.: Trajectory Perturbations of Fin-Stabilized Projectile Due to Muzzle Blast, J. Spacecraft and Rockets, 14(6), 1977, pp.339-347. [11] Schmidt, E. M., and Shear, D. P.: Optical Measurements of Muzzle Blast, AIAA Journal, 13, 1975, pp.1086-1096. 12) Ofengeim, D. K., and Drikakis, D.: Simulation of Blast Wave Propagation Over a Cylinder, Shockwaves, 7(5), 1997, pp.304-317. 13) Watanabe, R., Fujii, K., and Higashino, F.: Numerical Solutions of the Flow around a Projectile Passing through a Shock Wave, AIAA 95-1790-CP, 1995. 14) Ahmadikia, H. and Shirani, E.: Transonic and Supersonic Overtaking of a Projectile Preceding a Shock Wave, International Centre for Theoretical Physics Report-IC/2001/48, 2001. Nomenclature A: area, m2 a: speed of sound, m/s Cd: drag coefficient D: drag force, N 722 12