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Latin American applied research

versión impresa ISSN 0327-0793

Lat. Am. appl. res. v.38 n.3 Bahía Blanca jul. 2008

 

Numerical simulations of the flow around a spinning projectile in subsonic regime

J. Garibaldi1, M. Storti2, L. Battaglia2 and J. D'Elía2

1 Instituto de Investigaciones Científicas y Técnicas de las Fuerzas Armadas (CITEFA)
San Juan Bautista de La Salle 4397 (B1603ALO), Villa Martelli, ARGENTINA
ph.: (5411)-4709 8100, fx: (5411)-4709 8228
jgaribaldi@citefa.gov.ar, http://www.citefa.gov.ar

2 Centro Internacional de Métodos Computacionales en Ingeniería (CIMEC)
Instituto de Desarrollo Tecnológico para la Industria Química (INTEC)
Universidad Nacional del Litoral -CONICET
Güemes 3450, 3000-Santa Fe, ARGENTINA
ph.: (54342)-4511 594/5, fx: (54342)-4511 169
(mstorti, jdelia)@intec.unl.edu.ar, lbattaglia@ceride.gov.ar, http://www.cimec.org.ar

Abstract — The unsteady flow around a 155 mm projectile governed by the Navier-Stokes (NS) equations is numerically solved with a Large Eddy Simulation (LES) scheme, together with the SubGrid Scale (SGS) solved by a Smagorinsky model and the van Driest near-wall damping. The computed results are obtained in the subsonic flow regime for a viscous and incompressible Newtonian fluid in order to determine the axial drag coefficient, and they are validated against experimental data. The problem was solved by a monolithic finite element code for parallel computing on a Beowulf cluster.

Keywords — Spinning Projectile Model. Incompressible Subsonic Flow. Large Eddy Simulation (LES). Finite Element Method. Fluid Mechanics.

I. INTRODUCTION

There are two main factors to distinguish projectile aerodynamics from classic aerodynamics. The first one is the fact that most projectiles have an axis or plane of symmetry, which implies also symmetric aerodynamic parameters. The second one is related to large spinning velocities that, for tube artillery, are from 5,000 to 10,000 Revolutions per Minute (RPM), generating aerodynamic effects which are present only in the aeroballistic area. Among these particular parameters, there are the force and the torque produced by the Magnus effect. Although this force is relatively lower than the lift and can be ignored, the torque is critical for the projectile stability (Silton, 2002).

In general, a conventional artillery spinning projectile counts on a central cylindrical body to be guided through the cannon tube, and a frontal ogive whose length measures from one and a half to three times the caliber, see Fig. 1, resulting in a configuration with a drag higher than other flying corps, and where most of the drag force is due to the pressure. It is important to notice that this geometry places the pressure center ahead of the gravity center, see Fig. 2, that would make the projectile overturn if it were not stabilized. The mechanism used for avoiding the overturn is the spinning, i.e. the projectile is launched with an appropriate rotational velocity given around its longitudinal axis. This rotational velocity is generated during the travel of the object through the cannon tube due to the interaction between the rotating band and the rifling inside the cannon tube. Under the conditions described, any destabilizing torque over the projectile makes it react with a rotation over a plane normal to the torque plane, originating nutation and precession. The first of them is quickly damped, whereas the second holds during the rest of the trajectory, but it is restricted by design to 5° or less around the translational direction.

The flow around a projectile presents turbulent boundary layers, whose separation is a usual phenomena, and a large turbulent wake formed at the bottom of the object. In ballistic aerodynamics, prevention or control of the separation of the boundary layer is one of the most important aims, as well as an appropriate ogive design. Typical velocities for field artillery projectiles are from 150 m/s to 900 m/s or even more, reaching subsonic, transonic or supersonic flow regimes around them. Then, a complete characterization should consider appropriate analysis tools for each of them, as those employed in Sahu (1991) and Weinacht (2003).


Figure 1: Relevant projectile parts.


Figure 2: Sketch of aerodynamic forces acting on a projectile.

There is a considerable experience about the use of computational methods for solving flow problems around projectiles, most of them supported by validation against experimental results, for instance, in exterior ballistics (Sahu et al., 1999; Sahu and Heavey, 2004) as well as in the case of shell internal flows (Ray and Tezduyar, 2000).

As it is well known, a turbulent flow carries irregular and fluctuating fluid motions which significantly contribute to the transport phenomena. These flows are always three-dimensional, unsteady and mainly irregular except perhaps by large coherent macro structures, which are some kind of organized fluid motion that can be recognized in the instantaneous flow fields as well as in the time-averaged ones (Townsend, 1980). There are also eddies with a wide spectrum of sizes, from the larger ones close to the flow domain ones, to the much smaller ones at which viscous dissipation takes place. The numerical techniques available in Computational Fluid Dynamics to simulate them can be split in three main types (Pope, 2003; Davidson, 2004; Rodi, 2006; Sagaut, 2001):

  1. Direct Numerical Simulation (DNS): where a numerical method solves the three-dimensional Navier-Stokes and continuity equations (the NS system for brevity) for all scales and without any extra model for the smallest ones, so the mesh size must be smaller than the small-length scales. As the relation of this to the large-length scales varies inversely with the Reynolds (Re) number and the computation should be always three-dimensional, the number of grid points and computational cost increase roughly with the third power of the Re number, i.e., scale as O(Re3). Then, a DNS alternative results expensive for numerical studies in engineering applications considering that most of the computational effort is put over the smaller scales, which are not the important ones from the point of view of determining global flow values as forces over the projectile.
  2. Large Eddy Simulation (LES): is less expensive than DNS so it can be used in engineering cases where large-scale sizes dominate the flow behavior and steadiness prevails. Strictly, this type of numerical simulation is always three-dimensional and unsteady, where only the scale-motions greater than the mesh size are obtained by solving the unsteady Navier-Stokes system while the scale-motions smaller than the mesh size are accounted by some pair of subgrid and near-wall models. For high Reynolds number flows a near-wall model is more relevant than the subgrid one since the length-scales decrease with the increment of the Re number, in particular, the number of grid points needed for a good resolution of the near-wall zones increases roughly with the square of the Re number, i.e. scale as O(Re2). For comparison, it should be noticed that according to McDonough (1995), LES generally converges to DNS as the spatial and temporal steps decrease.
  3. Reynolds-Averaged Navier-Stokes (RANS): where all turbulent time-fluctuations are averaged out from the Navier-Stokes system, at least in the classical approachs. The time-averaged procedure introduces more unknowns than equations, giving the so-called closure problem. These new unknowns include the Reynolds stresses so they need a model for including also the whole spectrum of length scales, see Wilcox (1998); Mohammadi and Pironneau (1994). Even when there are many models for solving the Reynolds stresses, it is not easy to find the most appropriate, especially in the detached flow regions because they are not capable to deal with massively separated flows characterized by large coherent macro structures, which are determinant for the estimation of aerodynamic forces.

The Re number range to be considered indicates that the flow is fully turbulent. In this case, a LES model is adopted because it responds mainly to the computed large-scales, associated to the coherent structures developed due to the projectile motion. As already stated, the smaller scales are not solved but they are modeled, regarding that its influence over the other scales is related to energy transfers (Calo, 2005).

The projectile studied here is the PACU one (Proyectil Argentino de CUlote hueco), a 155 mm caliber shell designed by CITEFA and used now a days by the Argentinian Army. The aerodynamic study, which includes the spinning effects and the simulation of the turbulent flow regime with boundary layer separation is made by the PETSc-FEM code (Sonzogni et al., 2002; PETSc-FEM, 2007), which is a parallel multi-physics finite element program based on the Message Passing Interface (MPI) (Message Passing Interface (MPI), 2007) and the Portable Extensible Toolkit for Scientific Computations (PETSc) (Balay et al., 1997, 2005). Once the computer results are obtained, a comparison is made to experimental data obtained by analyzing the projectile velocity evolution registered by Doppler radar measurements, as is extensively considered in Garibaldi (2005).

II. STABILIZED FINITE ELEMENTS BY A SUPG-PSPG METHOD

In this section, specific characteristics about the finite element method (FEM) selected for solving the problem are given, such as the discretized equations and the finite elements kind itself, as well as the parallel implementation.

A. Navier-Stokes equations

The Navier-Stokes equations for a viscous and incompressible fluid flow are written as

∇ · u = 0; (1)

ρ(∂tu + u · ∇u) = ∇ · σ + f (2)

in Ω ×[0,T]. The position vector x = (x, y, z) is related to an Eulerian reference system, t is the time variable, Ω is the flow region, [0,T] is some time interval, f is the volumetric source term while u = (ux,uy,uz) and ρ are the velocity and density, respectively. The fluid stress tensor σ is decomposed into its isotropic -pI and deviatoric T parts

σ = -pI + T (3)

where p is the pressure and I is the identity tensor. As only Newtonian fluids with constant physical properties are considered, its deviatoric part T is related linearly to the strain rate tensor ε(u) with

(4)

where µ is the dynamic viscosity of the fluid and (...)T denotes the transpose. Dirichlet, Neumann and wall law boundary conditions are taken into account, respectively,

(5)

and the initial condition is a specified divergence-free velocity field u(x, 0) = u0 on the flow domain Ω.

B. The SUPG-PSPG formulation

The NS equations in the incompressible case present two important difficulties for the solution with finite elements. First, the character of these equations become highly advective dominant when the Re number increases. In addition, the incompressibility condition represents a constraint on the system. This has the drawback that only some combinations of interpolation spaces for velocity and pressure fields can be used with the Galerkin formulation, namely those that satisfy the so-called Ladyzhenskaya-Brezzi-Babuska condition. The advection and the incompressibility equations are stabilized with the Streamline Upwind Petrov Galerkin (SUPG) (Hughes and Brooks, 1979; Brooks and Hughes, 1982) and the Pressure Stabilizing Petrov Galerkin (PSPG) (Tezduyar, 1992; Tezduyar et al., 1992) stabilization terms, respectively. In this way, it is possible to use stable equal order interpolations. To enforce the satisfaction of the discrete continuity equation, an extra stabilization term is added, called Least Square Incompressibility Constraint (LSIC). The flow domain Ω is partitioned in E finite elements Ωe, with e = 1, 2,..., E, while the interpolation spaces and weighting functions are:

(6)

with , where d is the number of space dimensions and the Sobolev space

(7)

for all Ω eE, where L1 is the set of polynomials of first order while E is the set of elements. The combined SUPG/PSPG formulation for Eqs. (1-2) can be written as: find and such that:

(8)

for all and . In Eq. (8) three stabilization terms are added to the standard Galerkin formulation: the parameter δh corresponds to the SUPG stabilization, the εh parameter corresponds to the PSPG one, and finally the νLSIC parameter to enforce the incompressibility constraint. The first two terms are defined over different functional spaces and they can be written as δh=τSUPG(uh·∇)wh and εh=τPSPGρ-1qh, where

(9)

while and are the Reynolds number based on the element velocity uh and a global characteristic velocity U, respectively, that is,

(10)

The element size he is computed as

(11)

where wa is the function associated to node a, ne is the number of nodes connected to the element and s is the streamline oriented unit vector, while the element size is defined as the diameter of the sphere with the same element volume. Finally, the function z(Re) in Eqs. (9) is defined as

(12)

C. Large Eddy Simulation

In LES techniques, the momentum balance equations are solved with an "effective" kinematic viscosity νe = ν + νt, which is the sum of the molecular part calculated as ν = µ/ρ, plus a "turbulent" one νt. The last one is estimated in the PETSc-FEM (Sonzogni et al., 2002) code by means of the Smagorinsky (1963) sub-grid model coupled with the van Driest near-wall damping factor fν, and given by

(13)

in which CS is the Smagorinsky constant (CS ≈ 0.10 for flows in ducts) and is the trace of the strain rate ε(u). The van Driest near-wall damping factor fν reduces the "turbulent" kinematic viscosity close to the solid walls, but it introduces a non-local effect in the sense that the "turbulent" kinematic viscosity νt at a volume element also depends on the state of the fluid at the closest wall. A constant value A+ = 25 is adopted, while y+ = y/yw is the non-dimensional distance from the nearest wall expressed in wall units yw = ν/uτ, where uτ = (τw)1/2 is the local friction speed and τw is the local wall shear stress.

D. Parallel computing

The numerical simulations were performed using a domain decomposition technique (Paz and Storti, 2005) in the PETSc-FEM (Sonzogni et al., 2002) code, which is a parallel multiphysics finite element library (D'Elía et al., 2002; 2000; Storti and D'Elía, 2005) based on the Message Passing Interface (MPI, 2007) and the Portable Extensible Toolkit for Scientific Computations (PETSc) (Balay et al., 1997). The problem was solved using the Beowulf (2007) cluster Geronimo (2007), with 11 P4-nodes with 2 GBytes of RAM memory each. Other authors also appeal to parallel computations for solving similar problems, see Sahu et al. (1999); Sahu and Heavey (2004).

III. EXPERIMENTAL RESULTS

In Exterior Ballistics, the specialty related to projectile flight studies, the drag coefficient curves take a fundamental role for different applications, such as the generation of firing tables. These curves can be obtained also from tests inside wind tunnels or theoretical calculus. In the first case, the interferences between the projectile or its model and the tunnel walls affects the precision of results; in the second, the hypothesis adopted could move the results away from the real behavior.

Because of the limitations of the methods mentioned before, a different way for the making up of these curves is the identification of the aerodynamic properties of an object from flight tests over a real specimen. One of these techniques is the obtaining of drag curves from data registered by a Doppler radar, as it was made in this case. For this work, there were two devices: a TERMA DR-5000 trajectory analyzer, and also a rawinsonde Marwin MW12, which provides information to the radar about atmospheric conditions during the test flight.

The flight velocity data measured by the radar are used for determining the drag coefficient, called here CD. Notice that this coefficient does not correspond to a non-null attack angle nor a null one because this angle is an unknown under experimental conditions. Then, the CD calculated by the Doppler radar reflects the combined effects of the lift and drag induced by the gyroscopic angular movement of the projectile around its mass center.

The trajectory model used by the radar counts on three degrees of freedom, the same as in the widely used Point Mass Trajectory Model (PMTM) which is a model that allows the estimation of the trajectory and the impact points of projectiles. A more complete one is the Modified Point Mass Trajectory Model (MPMTM), that is applied when there are more aerodynamic coefficients available than in PMTM. Both of the methods are introduced in Calise and El-Shirbiny (2001).

On the other hand, a drag coefficient curve of an artillery projectile can be determined from radar measured velocity data obtained from flight tests through the Optimal Dynamic Fitting Method. A complete discussion about this subject can be found in Chen (1997).

Some shooting tests were carried out with PACU projectiles in proving grounds, under the scope of the CITEFA test projects and with the assistance of the equipment mentioned before, composed by a radar and a rawinsonde, where for the subsonic regime the drag coefficient estimated according to the measurements is CD ≈ 0.130.

IV. GEOMETRIC AND NUMERICAL MODEL

The geometry of the projectile was modeled by a computer assisted design (CAD) system, see Fig. 3, and then was inserted in a cylindrical cavity which conforms the whole domain of calculus, resembling a wind tunnel, as can be seen in Fig. 4.

The generation of an appropriate finite element mesh demanded more time than the expected one during the pre-processing stage because of the geometric definition of the model, and later for the quality of the finite elements obtained. The mesh finally used for calculation, see Fig. 5, was generated by a mesher developed in CIMEC (Calvo, 2005) and it counts on 756 k-elements of linear tetrahedral type and 159 k-nodes, approximately, where the space behind and around the projectile was carefully discretized considering the boundary layer, the wake and their influence over the results.

The input data were established based on the knowledge about artillery ordnance in CITEFA. From this point of view, the typical attack angle of an artillery projectile during a correctly stabilized flight and for a low-curvature trajectory is between 0o and 3o, as appreciated in the experimental tests. Higher attack angles are registered for trajectories with more curvature, as in the case of maximum reach, where the angles may take values of 5o or more in the apex. On the other hand, the translational velocities usually registered in subsonic flights are around 100 m/s when projection charges are of low velocity and short reach, whereas the rotational velocity is imposed by the rifling inside the cannon tube, as said before.

Regarding that, computational simulations over the projectile were performed for a translational velocity of 100 m/s, a spin velocity of 5200 RPM, and the kinematic viscosity of air of ν =1.5× 10-5 m2/s. These parameters allows the estimation of the Reynolds number, which takes values of order 1×107, indicating that the flow is turbulent and a turbulence model is needed for solving the problem. Besides, the flow is considered incompressible because the Mach number is M≈0.3, restricting the flow to the subsonic range.

V. NUMERICAL RESULTS

The calculations include the cases with attack angles of of 0o and 5o, being all of these conditions appropriate for comparing the numerical results with the experimental ones.


Figure 3: Main dimensions of the projectile model.


Figure 4: External mesh view.


Figure 5: Section of the domain of analysis, where mesh refinement can be appreciated in the wake and around the projectile.

The streamlines for null attack angle are shown in Fig. 6, where it can be seen that those originated closer to the object show a rotational effect. The vorticity over the projectile surface for the attack angles 0o and 5o are shown Fig. 7. In Fig. 8, the absolute velocity around the projectile is showed over a vertical plane for 0o of attack angle, while the static pressure near the projectile and over the wake are presented in Fig. 9 and the numerical drag coefficients for the same case can be seen in Fig. 10. Integration of the pressures registered over the projectile surface gave the net force exerted by the air on the object. These calculations are translated later to the drag coefficients in the three directions, CDx, CDy and CDz, which in the case of 5o take the mean corresponding values of 0.150, 0.098 and 0.430, see Fig. 11.


Figure 6: Streamlines for null attack angle. Those originated closer to the object show the rotational effect.


Figure 7: Vorticity over the projectile surface for the attack angles considered, 0o and 5o.


Figure 8: Absolute velocity around the projectile, showed over a vertical plane for 0o of attack angle.


Figure 9: Static pressure near the projectile and over the wake. Angle of attack of 0o.

VI. DISCUSSION

Comparing the drag coefficient curves obtained by the numerical simulations for 0o (Fig. 10) and 5o (Fig. 11), it is possible to see that in the second one there is a much higher lift in z direction and a more important lateral drag component in y direction due to the Magnus effect. The explanation is that the inclined projectile, which is spinning clockwise if it is seen from front, receives more wind underneath that produces a force which pushes it to left, i.e., to the +y axis, and an important increment in z direction.

Some comments arise after considering both the experimental and the numerical results, and they are summarized in the following paragraphs.

One of the most important issues is that the drag coefficient predicted by the PETSc-FEM code for an attack angle of 0o is , which has very good agreement with the one obtained experimentally, . Although the numerical calculations does not take into account the precession movement that is certainly present in shooting tests, this difference does not invalidate the computational analysis as a good first approximation. Besides, a correctly stabilized projectile over an almost rectilinear trajectory does not show attack angles higher than 3o.

The elaboration of a series of analogous numerical studies to be solved with PETSc-FEM involving velocities of 100 to 200 m/s and attack angles from 0o to 5o would give an even better characterization of the subsonic projectile flight.

As mentioned in Sec. III., there are trajectory methods for determining the impact points of the shell. Among them, the already mentioned MPMTM is able to predict the whole flight trajectory of a projectile and even the drift, i.e., the displacement of the object outside of the shooting plane. By applying this method, it could be possible to make another verifications, taking the drag, lift and Magnus coefficients from the numerical analysis, and making a comparison with the impact points experimentally determined by CITEFA with a Global Positioning System (GPS) in the artillery proving ground.

VII. CONCLUSIONS

Large Eddy Simulations of the unsteady flow around a spinning projectile model were performed. The flow around this body is dominated by large coherent macro structures formed by a massive flow separation on the rear end. There was also a good agreement between the experimental and numerical predictions, evidenced by the similarity between the drag coefficient determined experimentally and numerically for the studied range of parameters. Finally, it is concluded that, according to the available computational resources, LES may be used as a feasible numerical technique for high Reynolds number flows with detached flow regions in projectile aerodynamics.


Figure 10: Drag coefficients determined numerically for null attack angle.


Figure 11: Drag coefficients for 5o of attack angle.

ACKNOWLEDGMENTS
This work was performed with the Free Software Foundation GNU-Project resources as Linux OS and Octave, as well another Open Source resources as PETSc, MPICH and OpenDX, and supported through grants CONICET PIP-02552/2000, ANPCyT FONCyT (PME-209 Cluster, PID 99-74 Flags), ANPCyT PICT-6973-BID-1201-OC-AR Proa and CAI+D-UNL-2000-43. The first author is Mechanical Engineer and Technical Division Chief of the CITEFA Weapons Systems Department. The support and grant of time given by the Technology and Innovation Management Office of CITEFA is also acknowledged.

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Received: September 20, 2007.
Accepted: October 25, 2007.
Recommended by Subject Editor Eduardo Dvorkin.

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