MIRAMARE - TRIESTE June 2001

Transcription

1 IC/2001/48 United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS TRANSONIC AND SUPERSONIC OVERTAKING OF A PROJECTILE PRECEDING A SHOCK WAVE H. Ahmadikia Bou-Ali-Sina University, Hamadan, Iran and The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy and E. Shirani* Department of Mechanical Engineering, Isfahan University of Technology, Isfahan, Iran and The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. Abstract Transient flows with transonic and supersonic motions of a two-dimensional and axisymmetric projectile overtaking a moving shock wave are consider. The flow is simulated numerically by using full Navier-Stokes equations with Baldwin-Lomax algebraic turbulence model. The flow features which are very complicated and involve moving shock waves, expansion waves, boundary layers, wakes and their interactions are obtained and some flow parameters such as pressure distributions on the projectile surface and the drag coefficient are obtained. The results show that as the projectile passes through the moving shock wave, it changes the flow field features and pressure distribution dramatically. The drag force decreases and even become negative while the projectile takes over the shock wave. The change in the flow features and the aerodynamic forces in transonic flows are much more than the supersonic case, and the flow structures in two-dimensional is more complicated than in axisymmetric case. MIRAMARE - TRIESTE June 2001 * Regular Associate of the Abdus Salam ICTP. eshirani@cc.iut.ac.ir

2 1- Introduction The aerodynamic characteristic of flow, which is affected by the movement of a highspeed projectile overtaking a preceding shock wave from behind, is considered. This problem is a simplified version of a high-speed projectile, which passes through a highly under expanded supersonic exhaust plume produced by the projectile, while leaving a cylinder as shown in Fig.(l). When the projectile passes through the moving shock wave, it moves form a high-pressure region, which is the flow behind the moving shock, to the region of lower pressure in front at the shock wave. Such flows are observed near the outlet of a ballistic range facility where lunched projectile overtakes the performed blast movement. Examples of such problems are the thruster, retro-rocket firings, silo injections, shock tunnel discharges, launching tube fired projectile, etc. When a projectile overtakes the shock wave, complicated interaction of compression and expansion waves are produced. Thus projectile experiences drastic changes in the aerodynamic forces that are applied to it. These changes are very important in control and stability of the projectile. In fact, such a rapid change of aerodynamic forces may make the trajectory of the projectile unstable after the projectile overtakes the moving shock wave. Most of the studies in this subject are experimental observations and investigations [1,2]. Since this phenomenon takes place in a very short interval of time, the measurements of the aerodynamic forces are extremely difficult in this period. The diffraction of a shock wave from objects like cylinder and sphere are considered by many researchers, for example see [3], [4] and [5]. But in most cases, the object is stationary with respect to its surroundings. Wantanabe et al. [6] studied the subsonic movement of a projectile while overtakes a preceding shock wave from behind by a numerical technique. In this work, we investigate the movement of a projectile in a transonic and supersonic flow passing through a normal moving shock wave. In the transonic flows, the drag force is highly sensitive to the value of the flow Mach number. For example, it increases 100% when the free stream Mach number is increased from 0.95 to 0.97 [7]. To obtain such a vast changes in aerodynamic forces, one needs to simulate the transonic flow with reasonable accuracy. We use the full Navier-Stokes equations to simulate axisymmetric and two-dimensional flows around a projectile. The flow structures, pressure distributions and drag forces are obtained as a function of time as the projectile passes through the moving shock wave. Under expanded Jet Air Blast Fig. 1 Schematic of the flow features for a projectile passes through a self -produced plume 2- Flow Features Flow field, the projectile and the preceding shock wave are which are consider in this paper schematically shown in Fig. (2). In this figure, U s, U p and «2 are absolute velocities of the shock wave, projectile and flow behind the shock wave. If the projectile's absolute motion is supersonic, then depending on the value of relative

3 velocity of the projectile, its motion with respect the flow behind the shock can be subsonic, sonic or supersonic. Flow features relative to the projectile is shown in Fig. (3). In this figure, the projectile is assumed to be stationary and the relative velocities of the flow and shock wave are shown. Shock wave 1 Projectile UP-U s U P -u 2 Fig. 2 Flow field with the absolute velocities Fig. 3 Flow field with the relative velocities The relations between the flow properties for the moving shock wave are as follows [4]: p 2 Pi P 2 Pi ^ = r(p 2 / Pl )+l (2) (3) (4) where, T = (y + l)/(y -1) and c = (yp/p) u2. M s is absolute shock Mach number, p, p, y and c are the pressure, density, specific heat ratio and speed of sound, respectively. The states (1) and (2) are the locations ahead and behind of the moving shock wave, respectively. Assuming M p o is the absolute projectile Mach number and M P 2 is the relative projectile Mach number with respect to the flow behind the shock, then: fca M, (5) where, M p0 =U P /c i and M p2= (U P -u 2 )l c 2. Since M s is known, u 2 /u l and c 2 lc- [ can be obtained from equations (3) and (4). Although the projectile Mach number with respect to the ground, M p o, is greater than one, the flow over the projectile could be subsonic, M P 2<1 or supersonic, M P 2>1. Mach number relations behind the preceding shock wave and the region of subsonic or transonic overtaking are shown in Fig. (4). This figure is obtained by inserting, M P 2-\ in the equation (5). If M p o is above the solid line or between the lines, supersonic or subsonic overtaking will be taken place, respectively. If M p o is under the dashed line, the projectile will not take over the shock wave. In Fig. (5), M P 2 is plotted

4 against M p o for various shock Mach numbers. This plot is obtained by using equation (3) to (5). 3. Governing Equations The governing equations for two-dimensional and axisymmetric unsteady flows, the full Navier-Stokes equations, can be written as follows: where, = y" dx dx dy dx dy p pu pv e. 'pu pu 2 puv fe + + p p)u Fi=y" pv puv pv 2 +p (e, + p)v (6) = y H = -i)k- l/2p(m 2 +v 2 )J du ( du 0 0 m(p-t ge ) 0 u and v are velocity components, e t is total internal energy and T is temperature, m is equal to one for axisymmetric and is equal to zero for two-dimensional flows, [i, [i t, k and k t are molecular and turbulent viscosity and thermal conductivity, respectively. 4- Numerical Solution The governing equations are transferred into a body fitted coordinate system and are solved numerically. The Baldwin-Lomax [8] algebraic two-layer turbulence model is used. The equations of motion are linearized by using Newton approach. The convective fluxes are approximated by flux-splitting Roe's Reimann method [9] and [10], and the diffusion fluxes are approximated by the second order central difference scheme. The time derivatives are approximated by the second order backward difference. To prevent entropy deflection around the sonic lines, Harten and Hyman [11] entropy conditions are used. The minimum of two-gradient limiter is also employed to prevent numerical oscillations. For the two-dimensional flow, the whole

5 flow field is simulated and periodic boundary conditions are used at the adjusted points of the boundary. 5- Initial Conditions In order to specify the initial boundary conditions, the initial location of the moving shock wave must be chosen. Suitable initial position of the moving shock wave with respect to the projectile depends on the flow conditions. If subsonic or transonic overtaking is considered, then initial position of the incident shock must be located at far enough location from the projectile, so that the projectile affects on the flow can be calculated correctly. In this case, the CPU time required to solve the problem is very large, as will be indicated later. The initial conditions at the front of shock (state 1 on Fig. 2) are obtained according to equation (1) to (3). To obtain the initial flow field over the projectile, the steady flow around it is numerically solved by using the flow condition behind the shock as free stream conditions. 6- Grid Generation The accuracy and efficiency of numerical solution depend considerably on the quality of the grid employed. In this work, H-type structure grid is used. This grid is easy to produce and since the geometry of the projectile and flow domain is not complicated, it is suitable for this problem. The SOCBT 1 geometry is used for the projectile [12]. The number of grid points along the projectile for axisymmetric case is 300 and normal to it is 110. For the supersonic axisymmetric case, the solution domain is 40 percent less than the transonic case, because in this case the space in front of the body could be much less. For two-dimensional flows, the "O" grid is used. The number of grid nodes along the body is 400 and normal to the body is 110. In this case, solution domain is 30 percent greater than the axisymmetric case. 7- Results A computer code is developed and tested for several test cases to make sure that the code works properly, [13] and [14]. Also the diffraction of normal shock wave with a cylinder and sphere are simulated and compared with experimental data [5]. The shock Mach number, M s, for both two-dimensional and axisymmetric cases is assumed 2. According to equations (2) to (5), or Fig.'s (4) and (5), when 2<M p o<2.55 or M p o>2.55, the flow over the projectile is subsonic or supersonic, respectively. Two values of, M p o= 2.47 and 3.07 are chosen which corresponds to M P 2=0.94 and 1.4, respectively. The results for transient axisymmetric and two-dimensional cases are presented here. First the transonic axisymmetric flow is presented then supersonic axisymmetric flow and finally the results for two-dimensional case are presented. Fig. (6) shows the grid for transonic axisymmetric case. In this grid Ay mr >/(i=4.1xl0" 3 and ks m i n /d=5axl0~ 3 ' where d is the projectile diameter and s is arc length of the grid along the projectile surface. Fig.(7) shows the steady state solution which is needed as an initial condition for the axisymmetric transonic case, M P 2=0.94. The results are obtained for k-e and Baldwin-Lomax models. In this figure, the pressure is compared with experimental data [12] and [15] and good agreement is obtained. As it is shown both k-e and Baldwin-Lomax models give similar results. So the Baldwin-Lomax model is adequate to simulate the flow field and the rest of the results shown in this paper are obtained by using this model. Finally to the knowledge of the authors, there 1 - Secant-Ogive-Cylinder-Boat Tail

6 is not quantitative experimental results for the transient case to be compared with. So in this paper only the steady state solution is compared with the experimental data. Fig. (8) shows the sequence of the developments of the flow around the projectile overtaking the preceding shock wave, for transonic flow when M P 2=0.94. The density contours are shown at various time steps, in this figure. When the shock waves passes through the projectile leading edge, it is reflected, and the shock and reflected waves move along the projectile axes. As the shock waves move downstream, the angle between the two waves is decreased, until a Mach effect is formed. In this case a new shock wave, the Mach shock, is formed. The triple point, which is the location where three shock waves coincide, moves downstream and a slipstream and vortices are produced. As the incidence shock wave moves downstream, it interacts with boundary layers. Then the shock wave interacts with the expansion waves at the boat tail of the projectile and its shape is changed, Fig. (9). As shown in the figure, the base corner vortices are produced at the edge of the projectile back and move downstream. Also the shock wave interacts with the wake behind the body and complicated phenomenon is shaped. Finally, the shock wave moves downstream away from the body and eventually a steady state solution is obtained. During this period the flow structure around the body does not change much. Fig. (10), shows the density contours, at different time steps for M P 2=IA, which is when the supersonic overtaking is taken place. In this case, the moving shock wave considers with the oblique shock which is formed in front of the body due to the supersonic nature of the flow. The shock wave is affected as a result of such collision, and the oblique shock become closed to the body. The diffracted wave interacts with the expanded waves formed on the projectile surface, its tail edges. Then it interacts with the wake of the body and moves further downstream until the steady state solution is obtained. The flow in the wake region is shown in Fig. (11). Fig.'s (12) and (13) show the pressure distributions at the different time steps for the transonic and supersonic axisymmetric flows, respectively. In these figures, the pressure coefficient is defined as Cp = {plp 2 -l)l{0.5yml). As the projectile passing through the moving shock wave, it moves from high-pressure region to the low-pressure region. As a result of vortex-shock interactions, the pressure on the projectile surface changes considerably at the location of the shock wave. It is specially more important, when the flow is supersonic. The pressure distribution is unchanged after the shock waves moves downstream away from the body. The pressure becomes fixed after I.Its and 6.5ts for supersonic and transonic cases, respectively. Where ts=l/uioo and / is projectile length. The drag coefficient and effect of grid size for transition case is shown in Fig. (14). The drag coefficient is nearly constant before the projectile reaches the shock wave. But as soon as it reaches the shock waves, the drag force decreases suddenly and it becomes negative. In this case, not only the pressure drag is against the projectile movement, but also it accelerates the projectile. This is due to the sudden decrease of the pressure in the front part of the projectile that has passed through the shock wave. Even after the shock passes through the whole body, the drag force is still negative and it takes some time before the flow becomes steady and the effects of the shock wave in the flow field disappears. Due to the long CPU time needed to calculate such a problem, the results are not shown for longer time when the flow becomes steady. Fig. (15) shows the drag coefficient for supersonic overtaking, when M P 2-IA. In this case, the drag force is constant until the shock wave reaches the leading edge of the body as expected. As the shock wave moves along the projectile axis, the drag force is decreased to But after the shock wave leaves the projectile and its effects on the

7 flow field is vanished and the drag force increases again to reach its steady state value, which is In transonic flows, the change in drag coefficient is 1.3, which is larger than for supersonic flow. Also it takes more time for transonic flow to reach the steady state compared to supersonic case. The drag coefficient reaches 0.18 as the flow becomes steady. Effect of grid size on the drag coefficient is shown in Fig. (15). The number of grid points of the projectile grid is reduced from 300x110 to 225x75. The reduction of grid points has no effect on the qualitative behavior of the drag coefficient history. The drag coefficient for the two-dimensional transonic flow is shown in Fig. (16). In this case, the shock and vortices are stronger and therefore the change in drag coefficient is more than the previous cases. As the projectile moves toward the shock, the drag coefficient is nearly constant, but it suddenly decreases when the projectile reaches the shock. After the projectile passes through the shock, due to the vortex shedding on the projectile tail and their interaction with the shock, some changes in the drag coefficient are observed and finally as time goes on, the drag coefficient reaches its steady state value. It requires long CPU time to reach the steady state solutions, that is the reason why it is not shown in the figure. Fig. (17) shows the drag coefficients for supersonic overtaking of two-dimensional projectile. In this figure compared to the transonic case, the variation of the drag coefficient with respect to time is smaller and smoother. The reason is that in this case, the vortex shedding in the wake region is much less. On the contrary, the drag coefficient for two-dimensional flow does not reach a negative value as it was in the axisymmetric cases. 8- Conclusion It is shown that the numerical solution of Navier-Stokes using control volume approach and Reimann Roe flux splitting method with minimum of two gradient limiter and Harten-Hayman entropy condition is capable of simulating the transient motion of the flow with a projectile overtaking a moving shock for both transonic and supersonic overtaking and also two-dimensional and axisymmetric bodies. Complicated features of the flow are captured and the drag coefficient is shown to change drastically when the projectile passes through the moving shock. The effects of the overtaking on the flow features and the aerodynamic forces in transonic case are much more than the supersonic case. For axisymmetric flows, the drag force become negative and the projectile is pushed forward by the pressure drag. Acknowledgments This work was done at Bou-Ali-Sina University and the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. We appreciate the support of both organizations. This work was done within the framework of the Associateship Scheme of the Abdus Salam ICTP. References [1 ] Schmidt, E. M. and Shear, D. D., "Optical Measurements of Muzzle Blast," AIAA Journal, Vol. 13, No. 8, pp ,1975. [ 2] Erods, J. I. and Del Guidice, P. D., "Calculation of Muzzle Blast Flowfields," AIAA Journal, Vol. 13, No. 1, pp , [3] Bryson, A. E., Gross, R. W. F., "Diffraction of Strong Shocks by Cones, Cylinders and Spheres," Journal of Fluid Mechanics, Vol.10, No.l, pp. 1-16, 1961.

8 [4] Yang, J. Y., Liu, Y. and Lomax, H., "Computation of Shock Wave Reflection by Circular Cylinder," AIAA Journal, Vol. 25, No. 5, pp , [5] Shirani, E. and Ahmadikia, H., " The Diffraction of Shock Waves over a Cylinder and a Sphere Using Roe's Scheme," The Eighth Asian Congress of Fluid Mechanics, Shenzhen, pp China, [6] Wantanabe, P., Fujii, K. and Higashino, F., "Three Dimensional Flow Computation Around a Projectile Overtaking a Preceding Shock Wave," Journal of Spacecraft and Rockets, Vol. 35, No. 5, pp , [7] Nietubicz, C. J., "Navier-Stokes Computations for Conventional and Hollow Projectile Shapes at Transonic Velocities," AIAA Paper , Palo Alto Calif [8] Baldwin, B. S. and Lomax, H., "Thin Layer Approximation and Algebraic Model for Separated Turbulent Flow," AIAA Paper, , [9] Roe, P. L., "Characteristic-Based Schemes for the Euler Equations," Annual Review of Fluid Mechanics, Vol. 18, pp , [10] Roe, P. L., "The Use of the Reimann Problem in Finite Difference Schemes," Lecture Notes in Physics, Vol. 141, pp , [11] Harten, A. and Hyman, J. M., "Self Adjusting Grid Methods for One-Dimensional Hyperbolic Conservation Laws," Journal of Computational Physics, Vol. 50, pp , [12] Nietubicz, C. J., Inger, G. R. and Danberg, J. E., "A Theoretical and Experimental Investigation of a Transonic Projectile Flowfield," AIAA Journal, Vol.22, No. 1, pp , [13] Ahmadikia, H. and Shirani, E., "Assessment of Roe's Reimann Solver in Hypersonic 2-D and Axisymmetric Blunt Body Flows," Iran, Isfahan University of Technology, Esteghlal, Vol. 11, [14] Ahmadikia, H., Shirani, E, "Transonic Flow over a Projectile Using k-e Turbulence Models", Proceedings of The Third Aeronautical Engineering Conference, Tehran, Iran, 2000 [15] Cheing. C. C. and Danberg, J. E., "Navier-Stokes Computations Transonic Projectile at Angle of Attack and Comparison with Experimental Data," AIAA Paper, , 1988.

9 M, <po Ms Fig. 4 Projectile Mach number as a function of the shock Mach number Mp L! A^S 0 r i \ f ] ; i- ; \ ~ i ; " Ms-2.5 Ms=3.0 \ ; ^'jy\s,'\y\y\s-' - : yfyii-^i i! yy^\ \ \ : V/ y...ly'... *) ', ' ' ya Mpo Fig. 5 Relative projectile Mach number as a function of projectile Mach number a) b) Fig. 6 Grid configuration for transonic axisymmetric flows, a) The whole field, b) region around the projectile Exp. data 112] Exp. data 1151 Baldwin-Lomax model k- 8 model I xa I 0.8 I Fig. 7 Pressure distributions on the projectile surface for steady state case

10 t=0 (steady state) t=2.94 t=3.72 ^* t=3.96 t=4.66 t=o (steady state) Fig. 8 Flow features, density contours, as the projectile overtakes the shock wave, M p2 =0.94 r=4.07 Fig. 9 Flow features, density contours for M p2 =0.94 at the wake region 10

11 ^ ^ \ ""»W -OT-^*^^ * N t=0 (steady state) 1=1.24 t=1.86 f=2.22 t=2.61 t=oo (steady state) Fig. 10 Flow features, density contours, as the projectile overtakes the shock wave, M p2 =1.4 steady state t=1.86 t=2.21 t=2.61 Fig. 11 Flow features, density contours, for M P 2=1.4 at the wake region 0.6 :, - ^ /--^ 0.4 Cp i /1=3.36 i J \ Yr" j (=4.03 j 1 v ; W U=4.62 ' t > Cp \, ^ J t=0.38 l 1\J 1=0.67 I \ x/d r -f /! " - ; ; 1= t >1 2 Fig. 12 Pressure distribution for transonic axisymmetric case Fig. 13 Pressure distribution for supersonic axisymmetric case 11

12 0.4 c, ; li Grid-300*110 Grid-180*70.. i. i. i time Fig. 14 Drag coefficient as a functional time for transonic axisymmetric case c, D - VJ - Grid= 300*110 - Grid-210* time Fig. 15 Drag coefficient as a functional time for supersonic axisymmetric case c, 'CO.! - c, D Fig. 16 Drag coefficient as a functional time for transonic 2-D case Fig. 17 Drag coefficient as a functional time for supersonic 2-D case 12