Computational Ballistics III

COMPUTATIONAL

BALLISTICS III

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THIRD INTERNATIONAL CONFERENCE COMPUTATIONAL BALLISTICS

ON

COMPUTATIONAL BALLISTICS III

CONFERENCE CHAIRMEN C.A. Brebbia Wessex Institute of Technology, UK A.A. Motta Brazilian Navy Research Institute, Brazil

INTERNATIONAL SCIENTIFIC ADVISORY COMMITTEE C. E. Anderson R. Barauskas M. Beppu J. L. Drummond Alves D. Greatrix

N. Ishikawa Y. Kato S. Kubota W-S. Lee

ORGANISED

D. W. Pepper A. Peratta H. Rothe J. Sahu

BY

The Wessex Institute of Technology, UK

SPONSORED

BY

WIT Transactions on Modelling & Simulation

WIT Transactions on Modelling and Simulation Transactions Editor Carlos Brebbia Wessex Institute of Technology Ashurst Lodge, Ashurst Southampton SO40 7AA, UK Email: [email protected]

Editorial Board C Alessandri Universita di Ferrara Italy J Baish Bucknell University USA D E Beskos University of Patras Greece J A Bryant University of Exeter UK M A Celia Princeton University USA J J Connor Massachusetts Institute of Technology USA D F Cutler Royal Botanic Gardens UK G De Mey Ghent State University Belgium Q H Du Tsinghua University China A El-Zafrany Cranfield University UK S Finger Carnegie Mellon University USA M J Fritzler University of Calgary Canada G S Gipson Oklahoma State University USA

M A Atherton South Bank University UK C D Bertram The University of New South Wales Australia M Bonnet Ecole Polytechnique France M B Bush The University of Western Australia Australia A H-D Cheng University of Mississippi USA D E Cormack University of Toronto Canada E R de Arantes e Oliveira Insituto Superior Tecnico Portugal J Dominguez University of Seville Spain S Elghobashi University of California Irvine USA P Fedelinski Silesian Technical University Poland J I Frankel University of Tennessee USA L Gaul Universitat Stuttgart Germany S Grilli University of Rhode Island USA

K Hayami National Institute of Informatics Japan D B Ingham The University of Leeds UK D L Karabalis University of Patras Greece H Lui State Seismological Bureau Harbin China R A Meric Research Institute for Basic Sciences Turkey K Onishi Ibaraki University Japan M Predeleanu University Paris VI France S Rinaldi Politecnico di Milano Italy G Schmid Ruhr-Universitat Bochum Germany X Shixiong Fudan University China V Sladek Slovak Academy of Sciences Slovakia J Stasiek Technical University of Gdansk Poland M Tanaka Shinshu University Japan T Tran-Cong University of Southern Queensland Australia J F V Vincent The University of Bath UK Z-Y Yan Peking University China G Zharkova Institute of Theoretical and Applied Mechanics Russia

J A C Humphrey Bucknell University USA N Kamiya Nagoya University Japan J T Katsikadelis National Technical University of Athens Greece W J Mansur COPPE/UFRJ Brazil J Mikielewicz Polish Academy of Sciences Poland E L Ortiz Imperial College London UK D Qinghua Tsinghua University China T J Rudolphi Iowa State University USA A P S Selvadurai McGill University Canada P Skerget University of Maribor Slovenia T Speck Albert-Ludwigs-Universitaet Freiburg Germany S Syngellakis University of Southampton UK N Tosaka Nihon University Japan W S Venturini University of Sao Paulo Brazil J R Whiteman Brunel University UK K Yoshizato Hiroshima University Japan

COMPUTATIONAL

BALLISTICS III Editors C.A. Brebbia Wessex Institute of Technology, UK A.A. Motta Brazilian Navy Research Institute, Brazil

Editors: C.A. Brebbia Wessex Institute of Technology, UK A.A. Motta Brazilian Navy Research Institute, Brazil Published by WIT Press Ashurst Lodge, Ashurst, Southampton, SO40 7AA, UK Tel: 44 (0) 238 029 3223; Fax: 44 (0) 238 029 2853 E-Mail: [email protected] http://www.witpress.com For USA, Canada and Mexico WIT Press 25 Bridge Street, Billerica, MA 01821, USA Tel: 978 667 5841; Fax: 978 667 7582 E-Mail: [email protected] http://www.witpress.com British Library Cataloguing-in-Publication Data A Catalogue record for this book is available from the British Library. ISBN: 978-1-84564-077-4 ISSN: (print) 1746-4064 ISSN: (on-line) 1743-355X Library of Congress Catalog Card Number: The texts of the papers in this volume were set individually by the authors or under their supervision. Only minor corrections to the text may have been carried out by the publisher. No responsibility is assumed by the Publisher, the Editors and Authors for any injury and/ or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. © WIT Press 2007 Printed in Great Britain by Athenaeum Press Ltd. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the Publisher.

Preface The study of computational ballistics is one of the most challenging scientific endeavours, not only because of the diversity of time and space scales involved in practically all problems, but due to its interdisciplinary character. Interest in ballistics has also grown considerably recently as a result of the complexity of the political and military situation in many parts of the world. The classical subdivision into internal, external and terminal ballistics now encompasses numerous applications on how to model and protect against accidental or man made explosions. Many other civilian applications of ballistics are also of growing importance as impact, crash and blast problems are a by-product of more sophisticated techniques involving dimensions and speeds impossible to achieve a few years ago. The computational solution of problems in ballistics require specialised tools. Problems with very short time scales are particularly difficult to solve accurately as well as those involving changing of state or material and geometry conditions. Computational ballistics solutions frequently require substantial computer power and the use of specialised algorithms. The lack of sufficient number of open meetings in the field also preclude the sharing of important information amongst researchers and hence a way to avoid choosing the wrong alternatives. Because of that, conferences such as this are particularly important. This conference started in Rio in 2003 and was reconvened in Cordoba, Spain in 2005 before the meeting that took place in the New Forest. This volume covers the following topics: • Fluid flow aerodynamics • Interior ballistics • Terminal ballistics • Experimental mechanics/ballistics and field testing • New developments in computational techniques • Systems and technology The Editors are grateful to the members of the International Scientific Advisory Committee and other colleagues who helped to review the papers included in this book. They are also indebted to all authors for their presentations. The Editors New Forest, 2007

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Contents Section 1: Fluid flow aerodynamics Unsteady computational fluid dynamics modeling of free-flight projectile aerodynamics J. Sahu ...................................................................................................................3 3D transient numerical simulation for sabot separation D. Touati & S. Peles............................................................................................13 Fast calculation of direct fire trajectories taking the earth’s rotation into account W. Roetzel, W. Czarnetzki & T. Maier ................................................................21 Generating an aerodynamic model for projectile flight simulation using unsteady time accurate computational fluid dynamic results J. Kokes, M. Costello & J. Sahu..........................................................................31 Section 2: Interior ballistics Suppression of axial combustion instability in solid rocket motors D. R. Greatrix......................................................................................................57 ICE-ALE modelling for the combustion of homogeneous solid propellant A. Peratta & C. Peratta.......................................................................................67 Stochastic study of 60-mm gun-projectile responses M. Chen ...............................................................................................................77

Section 3: Terminal ballistics On the linkage of impact damage to modeling of ballistic performance J. M. Wells...........................................................................................................89 A method to increase the tip velocity of a shaped charge jet using a hollow cavity D. R. Scheffler & W. P. Walters..........................................................................99 A computational viscoelasticity/damage/plasticity model for high-explosive materials P. Rangaswamy & R. M. Hackett......................................................................109 Section 4: Experimental mechanics/ballistics and field testing Offshore platforms survivability to underwater explosions: part I A. A. Motta, E. A. P. Silva, N. F. F. Ebecken & T. A. Netto .............................123 Ballistic tests and numerical simulations for containment capability characterisation of Waspaloy® alloy M. Di Sciuva, M. Degiovanni, M. Mattone, M. Gherlone, N. Bonora, A. Ruggiero & C. Frola..................................................................131 Experimental survey of the pressure cone and temperature field of an A/T missile system K. Krallis, Ν. G. Orfanoudakis, M. Founti, D. T. Hountalas, A. Fatsis & N. Vlachakis ...................................................................................141 Section 5: New developments in computational techniques Safety checks and hit probability computation in numerical trajectory calculation using error propagation technology A. Kuhrt & H. Rothe..........................................................................................153 Scalable parallel computation of explosively formed penetrators (EFPs) E. L. Baker, C. Chin, B. E. Fuchs & J. Pham ...................................................163 Data mining in military systems A. A. Motta, A. S. Alves & N. F. F. Ebecken .....................................................171

Computer simulated fragmentation arena test for assessing lethality and safety separation distances of explosive fragmentation ammunitions V. M. Gold, E. L. Baker & J. M. Pincay ...........................................................181 Section 6: Systems and technology Evaluations of computational techniques for the engraving of projectiles J. South, B. Powers & M. Minnicino.................................................................193 Long range electromagnetic cannon E. Schmidt & J. Tzeng .......................................................................................203 Analysis of weapon systems protecting military camps against mortar fire M. Graswald, I. Shaydurov & H. Rothe............................................................213 Empirical relationship for muzzle exit pressure in a 155 mm gun tube D. Carlucci & J. Vega.......................................................................................225 Author Index ...................................................................................................231

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Section 1 Fluid flow aerodynamics

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Computational Ballistics III

3

Unsteady computational fluid dynamics modeling of free-flight projectile aerodynamics J. Sahu U.S. Army Research Laboratory, APG, MD, USA

Abstract This paper describes a new multidisciplinary computational study undertaken to model the flight trajectories and the free-flight aerodynamics of both a finned projectile at supersonic velocities and a spinning projectile at subsonic speeds with and without aerodynamic flow-control. Actual flight trajectories are computed using an advanced coupled computational fluid dynamics (CFD)/rigid body dynamics (RBD) technique. An advanced time-accurate Navier–Stokes computational technique has been used in CFD to compute the unsteady aerodynamics associated with the free-flight of the finned projectile at supersonic speeds and the spinning projectile at subsonic speeds. Computed positions and orientations of the projectile have been compared with actual data measured from free-flight tests and are found to be generally in good agreement. Predicted aerodynamics forces and moments compare well with the forces and moments used in the six degree freedom fits of the results of the same tests. Unsteady numerical results obtained from the coupled method show the flow field, the aerodynamic forces and moments, and the flight trajectories of the projectile. Keywords: unsteady CFD, projectile aerodynamics, couple CFD/RBD method, flight trajectory.

1

Introduction

Understanding the aerodynamics of projectiles, rockets, and missiles is critical to the design of stable configurations and contributes significantly to the overall performance of weapon systems [1-3]. The prediction of aerodynamic coefficients for these weapon systems is essential in assessing the performance of new designs. Numerical simulations have the potential of greatly reducing design costs while providing a detailed understanding of the complex WIT Transactions on Modelling and Simulation, Vol 45, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line) doi:10.2495/CBAL070011

4 Computational Ballistics III aerodynamics associated with each change. Recently, we have made progress in coupling computational fluid dynamics and flight dynamics to perform required multidisciplinary simulations for moving body problems. This involves realtime multidisciplinary-coupled computational fluid dynamics/rigid body aerodynamics computations for the entire flight trajectory of a complex guided projectile system. It can lead to accurate determination of aerodynamics that is critical to the low-cost development of new advanced guided projectiles, rockets, missiles, and smart munitions. Improved computer technology and state-of-the-art numerical procedures now enable solutions to complex, 3-D problems associated with projectile and missile aerodynamics. Modern guided munitions for future combat systems require the use of complex control surfaces (fins and canards), control mechanisms, and/or the use of flow technologies such as microjet gas generators to provide maneuver authority. The aerodynamic flow fields over these Army weapons are complex involving non-linear flow-physics especially during and after control maneuvers. For maneuvering munitions however, very limited data is available during and after control maneuvers [4,5], and there is a lack of knowledge and understanding of the associated unsteady aerodynamics. Accurate numerical modeling of this unsteady aerodynamics has been found to be challenging both in terms of time-accurate solution techniques and computing resources required. Our goal is to be able to perform time-accurate multidisciplinary-coupled computational fluid dynamics (CFD) and rigid body dynamics (RBD) computations for complex guided projectiles with control maneuvers using microjets and/or control surfaces such as fins/canards. As part of a DOD High Performance Computing Grand Challenge Project, the present work is focused on the coupling of CFD and rigid body dynamics (RBD) techniques for simultaneous prediction of the unsteady free-flight aerodynamics and the flight trajectory of projectiles. In other words, can we perform physics-based fly-outs of the projectiles on the supercomputers and accurately predict the unsteady aerodynamics and flight behavior of projectiles in actual flights?

2 Technical approach A real-time accurate approach is used in the present work; however, timeaccurate computations require much greater computer resources. The real-time accurate approach also requires that the six-degrees-of-freedom (6 DOF) body dynamics be computed at each repetition of the fluid flow solver. In threedimensional space, a rigid object has 6 DOF: three translations and three rotations. The 6 DOF code computes linear and angular velocities as well as the orientation of the missile, which are used as input to the computational fluid dynamics code. In turn, the aerodynamic forces and moments obtained from the flow solver are used to solve the 6 DOF body dynamics before moving on to the next time step. This procedure allows one to perform real-time multidisciplinary-coupled computational fluid dynamics/rigid body aerodynamics computations for the partial or entire flight trajectory of a complex guided projectile system. WIT Transactions on Modelling and Simulation, Vol 45, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

Computational Ballistics III

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The CFD capability used here solves the Navier-Stokes equations [6-9] and incorporates advanced boundary conditions and grid motion capabilities. The present numerical study is a big step forward and a direct extension of that research which now includes numerical simulation of the actual fight paths of the projectile using coupled CFD/RBD techniques using real-time accurate approach. The complete set of 3-D time-dependent Navier–Stokes equations is solved in a time-accurate manner for simulations of actual flights. The basic numerical framework in the code contains unified-grid, unified-physics, and unified-computing features. The user is referred to these references for details of the basic numerical framework. The 3-D time-dependent Reynolds-averaged Navier–Stokes (RANS) equations are solved using the finite volume method [7]:

∂ WdV + ∫ [F − G]⋅ dA = ∫ HdV ∂t V∫ V

(1)

where W is the vector of conservative variables, F and G are the inviscid and viscous flux vectors, respectively, H is the vector of source terms, V is the cell volume, and A is the surface area of the cell face. Second-order discretization was used for the flow variables and the turbulent viscosity equation. The turbulence closure is based on topology-parameter-free formulations. Two-equation [6] and higher order hybrid RANS/LES [10,11] turbulence models were used for the computation of turbulent flows. The hybrid RANS/LES approach is well suited to the simulation of unsteady flows and contains no additional empirical constants beyond those appearing in the original RANS and LES sub-grid models. With this method a regular RANS-type grid is used except in isolated flow regions where denser, LES-type mesh is used to resolve critical unsteady flow features. The hybrid model transitions smoothly between an LES calculation and a cubic k-ε model, depending on grid fineness. For computations of unsteady flow fields that are of interest here, dual timestepping as described below was used to achieve the desired time-accuracy [12]. An unique feature of the present coupled approach is the full grid motion capability that allows the grid to move translate and rotate as the projectile flies down the rage, since the grid velocity is assigned to each mesh point. To account for rigid body dynamics, the grid point velocities are set as if the grid is attached to the rigid body with 6 DOF. As shown schematically in Figure 1, the 6 DOF comprise of the three spatial coordinates (x,y,z) and the three Euler angles, roll, pitch, and yaw (φ, Φ, Ψ). For the rigid body dynamics, the coupling refers to the interaction between the aerodynamic forces/moments and the dynamic response of the projectile/body to these forces and moments. The forces and moments are computed every CFD time step and transferred to a 6 DOF module which computes the body’s response to the forces and moments. The response is converted into translational and rotational accelerations that are integrated to obtain translational and rotational velocities and integrated once more to obtain linear position and angular orientation. From the dynamic response, the grid point locations and grid point velocities are set. Both CFD and RBD computations are performed at every time step in a fully coupled manner. WIT Transactions on Modelling and Simulation, Vol 45, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

6 Computational Ballistics III

φ θ

Z

Y

Figure 1:

Rigid body dynamics schematic.

Figure 2:

3

X

ψ

Finned configuration.

Results

Time-accurate unsteady numerical computations were performed using Navier– Stokes and coupled 6 DOF methods to predict the flow field and aerodynamic coefficients, and the flight paths of both a finned projectile at supersonic speeds and a spinning projectile at subsonic speeds. In all cases, full 3D computations were performed and no symmetry was used. The supersonic projectile modeled in this study is an ogive-cylinder-finned configuration (see Figure 2). The length of the projectile is 121 mm and the diameter is 13 mm. The ogive nose is 98.6 mm long and the afterbody has a 22.3 mm, 2.5° boat-tail. Four fins are located on the back end of the projectile. Each fin is 22.3 mm long and 10.16 mm thick. The computational mesh for the 25-mm projectile model is an unstructured one. In general, most of the grid points are clustered in the afterbody fin region. The total number of grid points is about 4 million for the full grid. Here, the primary interest is in the development and application of coupled CFD and RBD techniques for accurate simulation of the free-flight aerodynamics and flight dynamics of the projectile in supersonic flight. The first step here was to obtain the steady state results for this projectile at a given initial supersonic velocity. Also imposed were the angular orientations at this stage. Corresponding converged steady state solution was then used as the starting condition along with the other initial conditions for the computation of coupled CFD/RBD runs. Numerical computations have been made for these cases at an initial velocity of 1034 m/s. The simulations were started a small distance away WIT Transactions on Modelling and Simulation, Vol 45, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

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from the muzzle. The corresponding initial angle of attack was, α = 4.9o and initial spin rate was 2500 rad/s. Fig. 3 shows the computed pressure contours at a given time or at a given location in the trajectory. It clearly shows the orientation of the body at that instant in time and the resulting asymmetric flow field due to the body at angle of attack. The orientation of the projectile of course changes from one instant in time to another as the projectile flies down range. Figure 4 shows the variation of the Euler pitch angle with distance traveled. As seen in this figure, both the amplitude and frequency in the Euler angle variation are predicted very well by the computed results and match extremely well with the data from the flight tests. One can also clearly see the amplitude damped out as the projectile flies down range i.e. with the increasing x-distance. Although not shown here, similar behavior is observed with the Euler yaw angle and it damps out with the increasing x-distance. The computed results again compare very well with the measured data from the flight tests. The time histories of the pitch and yaw angles are often customarily presented as a motion plot where the pitch angle is plotted versus the yaw angle during the flight of the projectile. It represents the path traversed by the nose of the projectile during the flight trajectory (looking forward from the back of the projectile). Such a plot is shown in Figure 5. This figure shows the comparison of the motion plots obtained both from the numerical simulations and the 6 DOF analysis of the flight results from ARFDAS [13]. Computed results match very well with the experimental flight test results. 8.00

Experimental Data CFD (2M grid) CFD (4M grid) CFD (6M grid)

6.00

T h eta, d eg

4.00 2.00 0.00 0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

-2.00 -4.00 -6.00 -8.00

Range, m

Figure 3:

Computed pressure contours.

Figure 4:

Euler pitch angle vs. range.

As stated earlier, coupled CFD/RBD simulations have been performed for a spinning projectile at subsonic speeds as well. The subsonic projectile is a 1.8caliber ogive-cylinder configuration (see Figure 6). A computational grid expanded near the vicinity of the projectile is shown in Figure 7. Grid points are clustered near the jet as well as the boundary layer regions to capture the high gradients flow regions. The computational grid has 211 points in the streamwise direction, 241 in the circumferential direction, and 80 in the normal direction. The unsteady simulations took thousands of hours of CPU time on the Xeon PC cluster system at ARL MSRC running with either 32 to 64 processors. WIT Transactions on Modelling and Simulation, Vol 45, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

8 Computational Ballistics III

(

8

(a)

)

y

6 4 2 0 -2 -4 -6 -8 -12

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

Psi

(b) Figure 5:

Motion plot (a) computation, (b) flight test. Lift

Microjet

Figure 6:

Spinning projectile geometry.

The first step here was to obtain the steady state results for the same projectile without the jet with the grid velocities prescribed. Also imposed were the angular orientations at this stage. Corresponding converged jet-off steady state solution was then used as the starting condition along with the other initial conditions for the computation of coupled CFD/RBD runs. Synthetic jets were activated at a specified point in the trajectory. The jet locations on the projectile are shown in Figure 6. The jet conditions were specified at the exit of the jet (sinusoidal variation in jet velocity). The jet conditions specified include the jet pressure, density and velocity components. The flow field inside of the tiny jet cavity is not computed. For the unsteady jets, time-dependent jet boundary conditions are applied at the jet exit. Numerical computations have been made for these jet cases at an initial Mach number, M = 0.39, initial angle of attack, α WIT Transactions on Modelling and Simulation, Vol 45, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

Computational Ballistics III

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= 2o, and an initial spin rate of 434 Hz. The jet width was 0.32 mm, the jet slot half-angle was 18o, and the peak jet velocity used was 110 m/s operating at a frequency of 1000 Hz. The jet-off coupled calculations are started from the same initial conditions. Z

Y Jet-on

Figure 7:

Computational grid.

Figure 8:

Schematic showing the jet actuation in a spin cycle).

22000.00

18000.00 ARFDAS Fit Experimental Data CFD

Ph i, d eg

14000.00

10000.00

6000.00

2000.00

-2000.00

0.0

10.0

20.0

30.0

40.0

50.0

60.0

Range, m

Figure 9:

Comparison of computed roll angle, jet-off.

Unsteady time-accurate CFD computations require huge computer resources. All the results presented here were obtained from unsteady numerical computations with a single synthetic jet on a 40–mm subsonic grenade (Figure 6). Looking from the back of the projectile (see Figure 8), the projectile spins clockwise. The jet actuation corresponds to one-fourth of the spin cycle from 45° to +45° with zero degree being the positive y-axis. The jet is off during the remaining three-fourths of the spin cycle. The unsteady CFD modeling technique required about 180 time steps to resolve a full spin cycle. The unsteady synthetic jet operates at a high frequency of 1000 Hz. For the part of the spin cycle when the jet is on, the jet operated for approximately four cycles. Trajectories were computed for both jet-off and jet-on conditions. For the jet-on case12, an unsteady synthetic jet was applied while the projectile is spinning and the jet was activated at approximately 41 meters into the trajectory. Figure 9 WIT Transactions on Modelling and Simulation, Vol 45, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

10 Computational Ballistics III shows the computed roll angles as a function of the x-distance or the range. The roll angles shown here are the accumulated values to include 360 degrees for every spin cycle of the projectile. Computed roll angle are compared with the experimentally observed values as well as data obtained the 6 DOF analysis of the flight results from ARFDAS [13]. and are found to be in good agreement. 8.00 7.00 6.00 5.00 4.00

Th eta-FP , d eg

3.00 2.00 1.00 0.00 -1.00

0.0

10.0

20.0

30.0

40.0

50.0

60.0

-2.00 -3.00 -4.00 -5.00

ARFDAS Fit Experimental Data CFD CFD(SM-DT)

-6.00 -7.00 -8.00

Range, m

Figure 10:

Computed Euler pitch angle, jet-off for different time steps.

Figure 10 shows the variation of the Euler pitch angle with distance traveled. As seen in this figure, both the amplitude and frequency in the Euler pitch angle variation are predicted very well by the computed results and match well with the data from the flight tests for up to a distance of 20 meters. For the rest of the trajectory, the frequency of the motion is predicted very well as seen by the comparison with the data, but the amplitude doesn’t compare as well with the data. Two sets of CFD results are shown in this figure, one obtained with the original time step and another one with a smaller time step reduced by half. As seen in this figure, reducing time-step by half has negligible effect on the accuracy of the computed results. Further analysis and investigation are required and are underway. Figure 11 shows similar behavior with Euler yaw angle with x-distance. The agreement between the computed results and the experimental data again is generally good except for the amplitude comparison at greater ranges. In both figures 10 and 11, we have both slow and fast mode frequencies and both compare well with the data and the ARFDAS fits. Figure 12 shows the variation of the Euler pitch angle with distance traveled both for jet-off and jeton conditions. As seen in this figure, the effect of the jet is small and incremental. The computed Euler angles are again compared with the experimental range test data shown in dark circles for the jet-off case. As seen here, the computed data predicts the frequency rather well; however, the amplitudes in some parts are under-predicted in the computations.

WIT Transactions on Modelling and Simulation, Vol 45, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

Computational Ballistics III

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8.00 7.00

ARFDAS Fit Experimental Data CFD (SM_DT) CFD

6.00 5.00 4.00 3.00

Psi-FP, deg

2.00 1.00 0.00 -1.00

0.0

10.0

20.0

30.0

40.0

50.0

60.0

-2.00 -3.00 -4.00 -5.00 -6.00 -7.00 -8.00

Range, m

Figure 11:

Comparison of the computed Euler yaw angle, jet-off for different time steps. 10.00 8.00 6.00

Theta (degree)

4.00 2.00 0.00 0.0

10.0

20.0

30.0

40.0

50.0

60.0

-2.00 -4.00 -6.00

Experimental Data CFD - Jetoff CFD - Jeton

-8.00 -10.00

Range (m)

Figure 12:

Comparison of computed Euler pitch angle, jet-off and jet-on.

4 Concluding remarks This paper describes a new coupled CFD/RBD computational study undertaken to determine the flight aerodynamics of both a finned projectile and a spinning projectile. A three-dimensional unsteady Navier–Stokes solver is employed to compute the time-accurate aerodynamics associated with the free-flight of the finned projectile at supersonic velocities and the spinning projectile at subsonic speeds. Computed positions and orientations of the projectile have been compared with actual data measured from free-flight tests and are found to be generally in good agreement. Predicted aerodynamics forces and moments also compare well with the forces and moments used in the six degree freedom fits of WIT Transactions on Modelling and Simulation, Vol 45, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

12 Computational Ballistics III the results of the same tests. This work demonstrates a coupled method to accurately predict the time-accurate unsteady aerodynamics and the flight trajectories of projectiles at various speeds. The present CFD/RBD simulations clearly show the capability of the coupled approach and form the basis for future multidisciplinary, time-dependent computations of advanced maneuvering munitions.

References [1]

[2]

[3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

Sahu, J., K. R. Heavey, and E. N. Ferry, “Computational Fluid Dynamics for Multiple Projectile Configurations.” Proceedings of the 3rd Overset Composite Grid and Solution Technology Symposium, Los Alamos, NM, October 1996. Sahu, J., K. R. Heavey, and C. J. Nietubicz, “Time-Dependent Navier– Stokes Computations for Submunitions in Relative Motion.” 6th International Symposium on Computational Fluid Dynamics, Lake Tahoe, NV, September 1995. Meakin, R. L., “Computations of the Unsteady Flow about a Generic Wing/Pylon/Finned-Store Configuration.” AIAA 92-4568-CP, August 1992. Smith B. L., and A. Glezer, “The Formation and Evolution of Synthetic Jets.” Journal of Physics of Fluids, vol. 10, No. 9, September 1998. Davis, S. A. and Glezer, A., “The Manipulation of Large- and SmallScales in Coaxial Jets using Synthetic Jet Actuators”, AIAA Paper No. 2000-0403, January 2000. Avancha, R. and Pletcher, R. H., “Large Eddy Simulation of the Turbulent Flow Past a Backward Facing Step”, AIAA paper No. 2000-0542, January 2000. Arunajatesan, S. and Sinha, N., “Towards Hybrid LES-RANS Computations of Cavity Flowfields”, AIAA Paper No. 2000-0401, January 2000. Batten, P., U. Goldberg and S. Chakravarthy, "Sub-grid Turbulence Modeling for Unsteady Flow with Acoustic Resonance", AIAA Paper 000473, 38th AIAA Aerospace Sciences Meeting, Reno, NV, January 2000. Peroomian, O., S. Chakravarthy, and U. Goldberg, “A ‘Grid-Transparent’ Methodology for CFD.” AIAA Paper 97-07245, 1997. Peroomian, O., S. Chakravarthy, S. Palaniswamy, and U. Goldberg, “Convergence Acceleration for Unified-Grid Formulation Using Preconditioned Implicit Relaxation.” AIAA Paper 98-0116, 1998. Goldberg, U. C., O. Peroomian, and S. Chakravarthy, “A Wall-DistanceFree K-E Model with Enhanced Near-Wall Treatment.” ASME Journal of Fluids Engineering, Vol. 120, pp. 457-462, 1998. Sahu, J., “Unsteady CFD Modeling of Aerodynamic Flow Control over a Spinning Body with Synthetic Jet.” AIAA Paper 2004-0747, Reno, NV, 58 January 2004. Arrow Tech Associates. “ARFDAS Technical Manual.” South Burlington, VT, 2001. WIT Transactions on Modelling and Simulation, Vol 45, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

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3D transient numerical simulation for sabot separation D. Touati & S. Peles Israel Military Industries, Central Laboratory Division, P.O.B. 1044, Ramat Hasharon, 47100, Israel

Abstract This paper presents a sabot discard model to predict sabots separation. The model uses the FSI (Fluid Structure Interaction) numerical packages of MSC-DYTRAN to determine the evolution of sabot separation motion. The 3D model enables us to calculate the asymmetric sabots discard trajectory using Euler Lagrange coupling. We neglect the blast region effect on the sabot discards process near the muzzle. Physical quantities such as linear and angular displacement, velocities, acceleration and exerted forces on the sabot structures are computed. Simulated results and experimental data show close agreement. Keywords: sabots petals, discard, kinetic energy penetrator.

1

Introduction

The kinetic energy penetrator is a widely used anti tank munitions. Its lethality is due to the kinetic energy imparted by the penetrator to the target with impact velocities of between 1.4 to 1.8 km/s. The penetrator is launched by means of a sabot assembly, consisting of three aluminum sabot petals, required to minimize the penetrator’s in bore balloting. The process of sabot separation begins as the projectile leaves the gun tube. Because of the transverse motion of the projectile within the gun, energy is stored in the elastic sabot petals. As the projectile leaves the muzzle, the constraints of the gun tube are released and the sabot elements are able to move laterally outward. Aerodynamic forces acting on each of the sabot petals causes them to lift up and disengage from the buttress grooves of the penetrator rod to permit its unconstrained, low drag flight to the target. It has been demonstrated by Schmidt and Shear [1] that aerodynamic interference generated by the sabot components can be a significant source of projectile WIT Transactions on Modelling and Simulation, Vol 45, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line) doi:10.2495/CBAL070021

14 Computational Ballistics III launch disturbance leading to unacceptable loss of accuracy at the target. Perturbations to the projectile trajectory are magnified by geometric asymmetry in the discard pattern and by the extended periods during launch when the sabot components are in close proximity to the projectile. Erengil [2] used analytical expressions to estimate transient sabot surface pressure distributions, which are then integrated numerically to yield force and moments to determine sabot discard trajectories. Cayzac et al [3] investigated the launch dynamics of a 44 mm APFSDS. The 2D Euler equations have been used to predict sabot separation in an intermediate ballistic flow field. Nusca [4] simulated a quasi steady sabot discard process that was also tested in a wind tunnel. First he computes the aerodynamic forces and moments exerted on the sabot petals at specific positions, and then this loading was used to compute the dynamic motion of the sabots. The quasi steady simulation ignores the flow time dynamics and does not link the aerodynamic forces to sabot motion. An insight into the transient sabots separation processes was provided in this paper via a detailed finite element model, representing the sabot and penetrator subjected to a free stream flow field. The model was prepared using exact geometric part features, which include the buttress grooves details on both the sabot and penetrator. We used the FSI numerical feature of MSC – DYTRAN, and utilize an Euler Lagrange coupling to correlate unsteady aerodynamic forces and the sabot body motion.

(a) Figure 1:

2

Computational domain

`

` (b)

Slice of the finite elements model of the sabot and projectile rod subjected to free stream flow field.

Numerical model

The configured numerical model included a stationary fins stabilized penetrator at a zero angle of attack with respect to the free stream flow field, and three 1200 sector sabots petals encompassing the penetrator. The sabot discard simulations are performed for a radial separation of the sabot from the projectile surface. Based on the above assumption, the model utilizes only one half of the computational domain as shown in figure 1a. The flow field characteristics were WIT Transactions on Modelling and Simulation, Vol 45, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

Computational Ballistics III

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calculated using the Euler equation and the angle of attack was used as the only source of asymmetry in the free flow stream initial condition. A slice of the finite elements model of sabot and penetrator, respectively, subjected to the free stream flow field is shown in Figure 1b. The numerical model neglected the blast region effect on the sabot discards process near the muzzle based upon experimental results which revealed a short range spatial influence of the blast region on the sabot discard process. The discard process comes to an end at a distance of 15 m from the muzzle exit while the blast region influence is limited to the close proximity of the muzzle at a distance of 1 m. The sabots and the kinetic projectile rod were modeled using 3 – Dimensional 8 – node Hexagonal elements. The sabot and the penetrator were assumed to be rigid. The interface at the buttress grooves between the sabot and the penetrator is modeled using contact features of the software to allow for realistic load transfer from the sabot to the penetrator. The interface between the penetrator and sabot buttress groove is shown in enlarged view in figure 2.

Figure 2:

Enlarged view of the sabot and projectile kinetic rod buttress groove interface.

The air surrounding the sabot and main rod structures is modeled using a Eulerian mesh with cylindrical outer shape border. Air inflow and outflow boundary conditions were applied and the air velocity at the boundary was the projectile muzzle velocity which was kept constant throughout the sabot discard simulation. An ideal gas equation of state was used to simulate the air behavior. The dynamic air pressure interacted with the sabots structure by means of a fluid – structure interacting algorithms.

3

Simulation results

Figure 3 shows the pressure contours of the air within close proximity of one of the sabot petals. WIT Transactions on Modelling and Simulation, Vol 45, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

16 Computational Ballistics III

(a)

(b)

(c) Figure 3:

(a) Shadowgraph photo of sabot discard experiment 3.5 ms after the projectile exits from bore muzzle. (b) Simulation results of air density plot in close proximity to the sabot region at 3.5 ms. (c) Contour map of density plots (simulation results).

Figure 4:

Contours of air pressure surrounding the sabot structure.

WIT Transactions on Modelling and Simulation, Vol 45, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

Computational Ballistics III

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The figure presents the sabot’s simulated and experimental attitudes with respect to the projectile rod at 6 m beyond the muzzle exit. There is a close resemblance between the simulated and experimental shock wave patterns. Figure 4 presents air pressure contours surrounding the sabot structure at different time frames during the discard process. Figure 5 depicts the linear velocities components of the sabot’s c.g during the discard process at (00 angle of attack).

Figure 5:

Figure 6:

Sabot C.G velocities components Vx (blue) and Vy (red).

Rotation angle vs. time of upper sabot at three different yaw angles.

Figure 6 shows the sabot’s angular displacement during the sabot discard process for different Yaw angles of the free flow. The measured rotation angle around the sabot’s c.g. point decreases as the Yaw angle increase. The angular WIT Transactions on Modelling and Simulation, Vol 45, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

18 Computational Ballistics III differences are due to Yaw angle effects which increase or decrease the petal area which is exposed to the flow field, causing the aerodynamic lift and drag force to discard the sabot at a slower or faster rate correspondingly. As may be seen, the interaction time between the sabot and the kinetic projectile rod does not exceed 2 ms for the specific sabot geometry and the given yaw angles. Figure 7 shows the aerodynamics force components exerted on the sabot during the discard process. The sabot separation process involves a continuous growth of the sabot surface area exposed to the free stream flow which results in an increase in the corresponding aerodynamic force magnitude in the x direction from 1 KN up to almost 10 KN while the resultant aerodynamic force magnitude in the y direction decreases.

Figure 7:

4

Aerodynamic forces exerted on sabot structure.

Summary

An insight into the transient sabots separation processes was provided in this paper via a detailed finite element model, representing the sabot and penetrator, subjected to a free stream flow field. The model was prepared using exact geometric part features, which include the buttress groove details on both the sabot and penetrator. It was written in the FSI numerical package of MSC - DYTRAN and utilizes an Euler Lagrange coupling to correlate the unsteady aerodynamic forces and the sabots’ rigid body motions. In retrospect we determined that the fluid Euler equation are sufficient for describing the sabot’s entire spatial motion and the general characteristics of the flow field over the sabot. The magnitude of the exerted aerodynamic interfacial forces were assessed, and it was further revealed that the interaction time between the sabot and the projectile rod does not exceed WIT Transactions on Modelling and Simulation, Vol 45, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

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2 ms for the specific sabot geometry. The most significant attribute of this model is that it does not rely on any empirical data and is therefore widely applicable. The simulation results are in close agreement with the experiments.

References [1] [2] [3]

[4]

Schmidt E.M., Shear D.D., Aerodynamic Interface During Sabot Discard, Journal of Spacecraft and Rockets, AIAA, Vol 15, No 3, May-June, 1978, pp. 235-240. Erengil M. E., Sabot Discard Model For Conventional and Electromagnetic Launch Package, 19th International Symposium of Ballistics 7-11 May 2001, pp 213-221 Interlaken, Switzerland. Cayzac R., Carette E., Alziary de Roquefort T. Intermediate Ballistics Unsteady Sabot Separation: First Computations and Validation, 19th International Symposium of Ballistics 7-11 May 2001, pp 297-305 Interlaken, Switzerland. Nusca M. J., Numerical Simulation of Sabot Discard Aerodynamics, Army Research Laboratory ARL-TR-204 September 1993.

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Computational Ballistics III

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Fast calculation of direct fire trajectories taking the earth’s rotation into account W. Roetzel1, W. Czarnetzki2 & T. Maier2 1

Helmut-Schmidt-Universität / Universität der Bundeswehr Hamburg, Germany 2 Hochschule Esslingen, Germany

Abstract An analytical approach is developed for the subsequent consideration of the Coriolis effect. The target is regarded as a moving target in the star-fixed coordinate system. The stationary atmosphere creates a nonuniform cross-wind which reduces the normal moving target deflection of the projectile. The approach is implemented in a previously developed analytical fast calculation method and tested against numerical calculations with good results. Keyword: Coriolis effect, analytical solution, power law, drag coefficient, Mach number.

1

Introduction

Several analytical solutions of the point mass equation of motion have been developed for the fast calculation of direct fire trajectories [1–4], which are based on the power law cD = C ⋅ (Ma )

−m

m

m

 a*   T  a = C ⋅   = C ⋅   ⋅  *  v  v  T 

m/2

(1)

for the drag coefficient variation with Mach number. McCoy [1] published flat fire solutions for m= 0, ½, 1. The later solutions [2–4] allow for arbitrary values of m and any angle of sight β [3, 4]. Wind can be considered using a coordinate transformation [2, 4]. For uphill and downhill firing the change of pressure and temperature along the trajectory should be considered. Usually mean values of WIT Transactions on Modelling and Simulation, Vol 45, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line) doi:10.2495/CBAL070031

22 Computational Ballistics III pressure and temperature are applied [3]. A more sophisticated way [4] is the correction of the Mach number exponent, which takes not only the changing pressure and temperature but also the curvature of the trajectory approximately into consideration. The above mentioned fast calculation methods do not allow for the earth’s rotation. In the present paper a simple analytical method is described [5] with which the Coriolis effect can subsequently be considered. The same coordinate system and nomenclature is used as in the foregoing paper [4].

2

Numerical method

For the precise calculation of the trajectory the complete equation of motion has to be integrated numerically. With the previously [2,4] defined ballistic coefficient D ( p, T ) = C ⋅

π d² 8M

ρ am = C ⋅

π d²

ρ * (a * ) ⋅ m

8M

p T*  ⋅  p *  T 

1− m / 2

(2)

G the equation of motion with wind velocity w can be written as [1] G G G G v = g − D ⋅ ( v − w

)

1− m

G G G ⋅ (v − w) − b

(3)

with the Coriolis acceleration G G G b = −2ω × v

(4)

G containing the earth’s vector ω of rotation G

 cosψ cos γ   . sinψ  − cosψ sin γ   

(5)

ω =ω

In eqn (5) the angle ψ is the latitude and the angle γ the azimuth of fire (x-axis), measured clockwise from north. Assuming the standard temperature drop with altitude ∆T/∆y = -0.0065 k/m [6] and regarding the atmosphere as perfect dry air yields the ballistic coefficient D as function of height y  0.0065  D = D0 ⋅ 1 − ⋅ y  T0  

4.25593+ m / 2

,

(6)

where index “0” indicates the firing site and origin of the earth-fixes coordinate system.

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G Starting at the origin with τ = 0 and L = {x, y, z} = 0 , the local projectile G velocity v can stepwise be calculated as function of time. Integrating G simultaneously over the time yields L as function of time. This numerical calculation is later carried out in order to test the new approach which is derived and described in the following.

3

Analytical approach

The equation of motion without the Coriolis acceleration is valid in a star-fixed not rotating system which moves with uniform velocity. Therefore the coordinate system in which the Coriolis-free solutions [1–4] are valid is considered to move G on uniformly with the fixed circumferential (not rotating) velocity u0 of the firing site (origin of coordinate system) at the instant of firing τ = 0 . For τ = 0 this star-fixed coordinate system coincides with the earth-fixed system, for τ > 0 they separate from each other. 3.1 The moving centre of gravity In this star-fixed system a flying projectile experiences an additional G gravitational acceleration ∆g in the horizontal direction, as the centre of gravity does not remain exactly perpendicular below the projectile. With the simplifying assumption of constant mean velocity components vx and vz and the radius R = 6.37·106m of the globe, the additional time dependent acceleration can be expressed as u x + v x  G g ⋅τ  g ⋅τ  ∆g = −  0 =− R  R  u z + v z 

− ω z ⋅ R + x1 / τ   . 0    ω ⋅ R + z /τ  1  x 

(7)

Substituting ωx and ωz according to eqn (5) and integrating twice over the time of flight yields the displacement of the projectile ω cosψ sin γ + x1 / Rτ 1  G g 3 . 0 ∆L1g = − ⋅ τ 1   6 ω cosψ cos γ + z / Rτ  1 1 

(8)

The vertical components ∆gy = 0 and ∆L1gy = 0 since cos ωτ 1 ≈ 1 . The derivation of eqn (8) is correct under vacuum conditions. The effect of eqn (8) is very week in normal cases and a sufficiently accurate approach. 3.2 The moving target The main effect of the earth’s rotation is the fact that in the uniformly moving star-fixed system the earth-fixed target appears as a moving target. We look first at the poles of the globe. There the circumferential velocity is zero, but the earthWIT Transactions on Modelling and Simulation, Vol 45, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

24 Computational Ballistics III fixed coordinate system rotates relatively to the considered star-fixed system. At the north pole any target will travel from west to east, i.e. from right to left. A negative angle of lead α < 0 would be required to hit the moving target. The G shift ∆L1ω of the target during the time of flight τ 1 can be expressed as

[ (

G G G G ∆L1ω = τ 1 ω × L1 + ∆L1g



G 



) ] =τ (ωG × L ) + ωG× ∆ L 

 1

G

1



(9)

1g





≈0

which leads with eqn (5) to z1 sinψ + y1 cosψ sin γ   G .  ∆L1ω = ω ⋅ τ 1 − x1 cosψ sin γ − z1 cosψ cos γ    y cosψ cos γ − x sinψ 1 1  

(10)

At other latitudes ψ ≠ ±90° an additional target movement takes place in the star-fixed system which is caused by the growing distance between the origins of both coordinate systems. However, a detailed analysis shows that this additional shift can be omitted. Its effect is compensated by the fact that in the usual standard free-fall acceleration g = 9.80665 m/s2 (at sea level) the centrifugal acceleration due to the earth’s rotation is included (subtracted from the mass attraction force). Thus, eqn (10) is valid for all values − 90° < ψ < +90° . The same shift (10) with negative sign is found by the twofold integration of the Coriolis acceleration eqns (4,5) using a constant mean projectile velocity G G v = L1 / τ 1 . The negative sign shows that in the earth-fixed system the projectile is apparently accelerated and displaced in the opposite direction as the target is in the star-fixed system. One could suppose that the final Coriolis deflection of the projectile in the earth-fixed system could be expressed as G G G ∆L1 = ∆L1g − ∆L1ω .

(11)

However, this is only a rough approximation under normal shooting conditions which will be discussed later in this paper. Eq (11) holds true in a vacuum. The G contribution ∆L1g is normally relatively small. In the special case of the G G downward free-fall in a vacuum at the equator ψ = 0, L1 = {0,− L1 ,0} ∆L1g is

(

)

decisive. For γ = 0 and with the time of flight τ 1 = (2 L1 / g ) correct [7, p.214] east drift

1/ 2

1/ 2

 2L  ∆L1z = ω ⋅ L1 ⋅  1   g 

1/ 2

 2L   1  ⋅  − + 1 = ω ⋅ L1 ⋅  1   3   g 

eqn (11) yields the

2 ⋅ . 3

(12)

G The term –1/3 is due to ∆L1g . Neglecting it would cause an overestimation of 50%. WIT Transactions on Modelling and Simulation, Vol 45, © 2007 WIT Press www.witpress.com, ISSN 1743-355X (on-line)

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3.3 The Coriolis wind The moving target problem under consideration differs from the usual case in so far as the air follows the moving target. If no wind is present, the stationary air in the earth-fixed system rotates in the star-fixed system together with the target. The moving target gets tailwind of equal velocity. The wind is not uniformly G distributed but the cross-wind velocity wc is zero at the firing site and grows G linearly with the distance to its maximum value w1 at the target G G x G x − ∆L1 wc = ⋅ w 1 = ⋅ . x1 x1 τ 1

(13)

This Coriolis wind pushes the projectile towards the moving target and reduces the deflection according to eqn (11). Since the consideration of uniform wind is no problem, a suitable mean uniform Coriolis wind velocity wG cm is defined and derived which produces the same effect as the actual linearly growing Coriolis wind does: G G G ∆L1ω − ∆L1g . G G ∆L1 f f f w cm = ⋅ w 1 = − ⋅ == ⋅

τ1

τ1

(14)

A horizontal flat shot in the x-direction (vx = v) towards a target at distance x1 with cross-wind of velocity wz = wc is considered. Gravity is neglected. For wZ