[American Institute of Aeronautics and Astronautics 29th AIAA Applied Aerodynamics Conference -...

American Institute of Aeronautics and Astronautics

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Navier-Stokes Predictions of Aerodynamic Coefficients and Dynamic Derivatives of a 0.50-cal Projectile

Sidra I. Silton* U.S. Army Research Laboratory, Aberdeen Proving Ground, Maryland, 21005

A computational study has been undertaken to predict the static aerodynamic coefficients and dynamic derivatives of the standard 0.50-cal army projectile using multiple simulations that are collectively called the virtual wind tunnel technique. Computational solutions have been completed and validated against experimental range data for a wide range of Mach numbers to include subsonic, transonic, and supersonic flight regimes. It was found that steady-state methodologies can be used to obtain the static aerodynamic coefficients and most of the dynamic derivatives with good agreement. However, the accurate prediction of the Magnus moment coefficient in the transonic and subsonic flight regimes remains problematic. Further investigation of grid resolution, including refined wake region, boundary layer growth rate and circumferential resolution, for the steady-state simulations, time-accurate, moving mesh RANS simulations, and LNS simulations have been completed without significant improvement seen in the prediction of the Magnus moment coefficient. The discrepancy in Magnus moment coefficient prediction is large enough that it appears to affect the predicted flight dynamics of the projectile.

Nomenclature

plC = roll damping coefficient

Cm = pitching moment coefficient

mm CCq = pitch damping moment coefficient sum

mC = pitching moment coefficient derivative

NC = normal force coefficient derivative

Cn = side moment coefficient

pnC = Magnus moment coefficient derivative

nC = slope of side moment coefficient with coning rate

d, D = projectile diameter, m Ix = projectile axial moment of inertia, kg-m2 Iy = projectile transverse moment of inertia, kg-m2 k = turbulence kinetic energy, m2/s2

k = reduced frequency m = projectile mass, kg M = Mach number

n = normalization factor for stability equations,

p = projectile spin rate (in non-rolling coordinate frame), radians/sec P = pressure, N/m2 qo = pitch rate, radians/sec R = undamped eddy viscosity, m2/s S = projectile cross-sectional area, m2

Sg = gyroscopic stability factor t = time, seconds

* Aerospace Engineer, Weapons & Materials Research Directorate, RDRL-WML-E, Senior Member.

29th AIAA Applied Aerodynamics Conference27 - 30 June 2011, Honolulu, Hawaii

AIAA 2011-3030

This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.


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T = time period, seconds V = free stream velocity, m/s xcg = center of gravity location = vertical component of angle of attack in non-rolling coordinates mean = mean angle of attack = angular amplitude, radians

t = total angle of attack, 22

= angular rate of change at , radians/sec = horizontal component of angle of attack in non-rolling coordinates = cosine of total angle of attack = sine of total angle of attack = turbulent dissipation rate, m2/s3

F,S = fast and slow mode damping exponents = density, kg/m3 = coning rate of projectile, radians/s

= angular velocity of projectile resulting purely from rotation about velocity vector

I. Introduction OMPUTATIONAL fluid dynamic (CFD) calculations are routinely used to accurately predict the static aerodynamic coefficients (e.g. drag, longitudinal, and lateral forces and moments) and flow phenomena of

many geometrically complex projectiles and missiles. The prediction of dynamic coefficients such as pitch damping, roll damping, and Magnus moment has not achieved as widespread use. Projectile designers still turn to flight tests (aeroballistic range and/or telemetered free-flight) to determine the full set of static and dynamic aerodynamic data needed to predict in-flight motion and projectile stability. However, computational resources are available today for predicting both static and dynamic coefficients, and therefore full projectile performance, stability, and free-flight motion of most projectiles and missiles.1

The 0.50-cal projectile has been in use for almost 60 years. While a limited amount of aerodynamic data was collected during the projectile’s development and testing2, a relatively large amount of aerodynamic data has been obtained over the years for these rounds. Most of the experimental data has been collected within the last 20 years at the former Ballistic Research Laboratory’s Free Flight Aerodynamics Range (BRL Aerodynamics Range).3,4 The results in Ref. 3 are the original linear reductions completed by McCoy. The data obtained from that test was reduced a second time using a non-linear data reduction technique that was not available at the time of the original test.4 More recently, experimental data has been collected on a more limited range of Mach number at the Defence Research and Development Canada Valcartier’s Aeroballistic Range.5 Previous computational results have been obtained for this or similar 0.50-cal round using Parabolized Navier-Stokes6 and Reynold’s Averaged Stokes7 (RANS) simulations with limited success. While both studies were able to predict the static aerodynamic coefficients, neither study presented a comprehensive look at the dynamic derivatives.

The present study is a comprehensive presentation of the static aerodynamic coefficients and dynamic derivatives of the 0.50-cal projectile over its flight from Mach 2.7 down to Mach 0.6. The previous RANS study on this projectile found that while steady-state simulations were able to accurately predict the Magnus moment coefficient in the supersonic regime, the agreement was not good in the transonic and subsonic regimes.7 More recent work by Army Research Laboratory (ARL) investigators also indicates that prediction of Magnus moment via steady-state methods in the subsonic and low-transonic regime may not be as accurate as those in the supersonic regime, at least for some projectile base shapes.8, 9 Time-accurate CFD with advanced turbulence modeling [e.g. hybrid RANS and large-eddy simulation (LES)] showed improved Magnus moment predictions.9, 10 This time-accurate methodology is applied here with limited success. Other possible methods to improve prediction of the Magnus moment, including grid resolution, other turbulence models, and time-accurate RANS moving mesh, are also investigated. The roll damping coefficient is typically predicted with reasonable accuracy via steady-state computations. Finally, the determination of the pitch-damping moment coefficient is also necessary for a complete picture. Both a steady-state, rotating reference frame methodology11, 12 and a time-accurate moving-mesh methodology12 is used to obtain this final dynamic derivative.

C


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Figure 2. Plane cut of 7.5 million hexahedral cellmesh (projectile in blue).

II. Numerical Approach

A. Computational Geometry and Mesh The computational model of the 0.50 cal. (1 cal. = 12.95 mm) projectile, including the outer boundary was

generated using Solidworks13. The projectile (Figure 1) is 4.46 cal. long, with a groove 0.16 cal. long by 0.02 cal. deep, with a 9° filleted boat tail. The center of gravity location, xcg, is 2.68 cal. from the projectile nose. The projectile spin rates were determined from the muzzle exit twist rate reported in Ref. 3: 2.94 cal. per revolution,

which gives a non-dimensional spin rate ( of 0.107. A full, three-dimensional (3-D) mesh was required in order

to simulate the spinning shell at angle of attack. Multiple meshes were used in this study. The initial mesh used was a 3-D hexahedral mesh created using

ICEMCFD14 (Figure 2). The outer boundary was created as a frustum with a 7° outward draft far enough from the body to minimize any possible interference – 9.3 body lengths from the nose, 12.5 body lengths from the base, and a minimum of 15 body lengths radially out from the body. The mesh contained 160 circumferential cells. In generating the mesh, boundary layer mesh spacing was used near the projectile body. Resolving the viscous boundary layer is critical for predicting the Magnus and roll damping moments. Therefore, the option to integrate the equations to the wall was utilized almost everywhere (not the groove sidewalls). Rather than the normal y+ value on the order of 1.0 required to adequately resolve the boundary layer, a y+ value of 0.5 or less was desired to ensure that the Magnus effect of the spinning shell was properly captured. In order to achieve this y+ value for all Mach numbers up to Mach 2.7, a first radial cell spacing of 0.5 m was chosen. All mesh stretching ratios were kept to 1.2 or less. Initially, the author planned to use a single mesh for both the RANS and the hybrid RANS/LES (LNS) simulations. As such, special care was taken to create nearly isotropic cells in the base region (Figure 3). Specifically, the stretching was kept to 1.01 in the near wake region (approximately 1.5 body lengths behind the projectile). The resulting mesh contained approximately 7.5 million hexahedral cells (RANS I mesh).

As the simulations progressed, a more optimal LNS mesh was desired. Therefore, the ICEMCFD mesh was modified to create separate meshes to be used for the RANS simulations and the LNS simulations. For the RANS only mesh (RANS II mesh), the stretching in the wake region was increased to 1.1. The remainder of the mesh remained approximately the same, which reduced the mesh size to approximately 5.5 million cells. For the LNS mesh, the wake region of the RANS mesh was removed and replaced with an LES mesh (even spacing). A non-conformal

Figure 1. Sketch of projectile (left) and 3-D representation of solid model (right)


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(a) (b)

Figure 4. Unstructured computational mesh generated using GRIDGEN: a) subsonic mesh and b) close-up ofsupersonic mesh around which subsonic mesh was built.

Figure 3. Close-up of mesh near projectile bodyshowing boundary layer spacing and isotropiccells in base region.

interface was created, which was handled within the CFD code. The resulting mesh size was 9.6 million cells.

For comparison, a second set of meshes (one for supersonic flow (M > 1.5) and one for trans- and subsonic (M ≤ 1.5)) was created using GRIDGEN version 15.15 This mesh contained hexahedral, pyramid, and tetrahedral cell types (Figure 4). These meshes were generated with a highly resolved wake region, but remain about the same size as the other mesh due to the use of tetrahedral cells. The near-body region of the meshes was similar, as the subsonic mesh generated by making an additional tetrahedral mesh around the supersonic mesh. Each mesh contained 144 circumferential cells and an o-grid was generated around the body, 0.57 cal. away, using the hyperbolic extrude feature with GRIDGEN. However, the first boundary-layer edge, which was 1.9 x 10-5 cal. for the supersonic mesh, was relaxed to 5.6 x 10-5 cal. for the subsonic mesh. These cell spacings ensured a y+ distribution less than one-half for all Mach numbers. The outer boundary for the supersonic mesh was located, 5.3 cal. radially, 5.8 cal. in front and 14.9 cal. behind for a cell count of approximately 5.9 million cells. The subsonic mesh had an increased cells count of 7.2 million cells with the boundary located at 89.9 cal radially, 90.3 cal in front and 89.4 cal behind the body.

A final set of meshes was created based on the RANS II, ICEMCFD mesh. Using the same base mesh, the circumferential resolution was increased to 240 cells and then 320 cells. This resulted in mesh sizes of 8.45 million cells and 12.0 million cells, respectively. The grids are not exactly 1.5 times and 2 times the original mesh as additional cells had to be added to ensure smooth transitions. The final mesh investigated reverted to the baseline 160 circumferential cell mesh. While the first cell spacing remained the same, the growth rate within the boundary layer was reduced to 1.05. The boundary layer mesh was further refined by approximating the required thickness from previous simulation results and ensuring a sufficient number of cells in the region.

B. Navier-Stokes CFD The commercially available CFD++ code, version 7.1.1 and version 10.1.1, was used in this study.16 The CFD++

code can simulate a range of fluid dynamic phenomena, ranging from incompressible to hypersonic flow. The 3-D, time-dependent, RANS equations are solved using the finite volume method. The spatial discretization was a second-order, multi-dimensional total variation diminishing (TVD) polynomial interpolation scheme. Solutions to semi-infinite “Riemann problems” are used in CFD++ to provide upwind flux information to the underlying transport


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scheme. Approximate Riemann solvers are used to determine the higher order fluxes to avoid spurious oscillations that may become physically unbounded if determined via fixed-stencil interpolation. A characteristics-based inflow/outflow boundary condition was used on all far field boundaries. This boundary condition takes a specified set of free-stream data, using it as a virtual state outside the domain, and solves a Riemann problem at the boundary. Far-field absorbing layers were used on the far field boundaries (not including outflow) for the transonic and high subsonic simulations. Additionally, the preconditioned equations were chosen at the lowest Mach numbers. For a majority of the present study, the steady-state, double precision, point-implicit scheme was utilized. The maximum Courant-Friedriech-Lewy (CFL) number was chosen using the recommendations within the CFD++ solver. For some of the steady-state simulations, it was necessary to reduce the maximum CFL number in order to obtain a converged solution. For the time-accurate simulations (both RANS and LNS) that were completed, the implicit solver with dual time-stepping was used.

Multiple turbulence models were used for the steady-state computation – the three-equation k--R model, the two-equation realizable k- model, and the two-equation cubic k- model. The three-equation k--R model solves transport equations for the turbulence kinetic energy, k, its dissipation rate, , and the undamped eddy viscosity, R. This model was previously found to provide the best overall performance for one configuration across the Mach number range.9 The two-equation realizable k- model solves the transport equations for k and and was previously used with reasonable results.7 The two-equation cubic k- model also solves the transport equations for k and while enforcing realizability and includes non-linear terms to account for nomal-stress anisotropoy, swirl, and curvature effects. For a majority of the grid resolution study, the two-equation realizable k- model was implemented. CFD++

also has hybrid RANS/LES models available. The hybrid RANS/LES approach using the Batten-Goldberg model (LNS) was used in this study. The LNS methodology of CFD++ is based on the solution of transport equations for the unresolved turbulence kinetic energy and its dissipation rate and incorporates anisotropy and low Reynolds number damping effects in both the LES and RANS modes. CFD++ reverts to the cubic k- model where the mesh is of a RANS-type and blends automatically to an anisotropic form of the Smagorinsky model in regions of uniformly-refined mesh (i.e. LES-type).16

Most of the simulations were performed in parallel on an 1100-node, 4400-core, Linux Networx Advanced Technology Cluster at the ARL Distributed Shared Resource Center (DSRC). Each node has two dual-core 3.0 GHz Intel Woodcrest processors. Some simulations were performed on an older 1024-node, Linux Networx Evolocity II system with two 3.6 GHz Intel Xeon EM64T processors per node. The number of processors used for each simulation was such that approximately 115,000 – 150,000 cells were partitioned on each processor, with fewer cells per processor for the older machine in order to achieve similar CPU time per time step. The steady-state calculations took approximately 12 – 15 seconds of CPU time per iteration and convergence was achieved in less than 1000 iterations. The solution was considered converged when the flow residuals had reduced at least 4 orders of magnitude and the aerodynamic coefficients change by less than 0.5% over the last 200 iterations, with the aerodynamic coefficients being the determining factor in all cases. The time-accurate simulations took approximately 45 seconds of CPU time per time-step.

The following sections describe the computational methodologies that were used to determine the dynamic derivatives, Magnus moment coefficient and pitch-damping moment coefficient, including boundary conditions imposed.

1. Magnus Moment Calculation Multiple turbulence models were investigated and multiple meshes were generated in order to determine how to

best calculate the non-linear Magnus moment. For the steady-state calculations, an isothermal, rotating wall boundary condition was imposed to simulate the spinning of the projectile. Time-accurate, moving mesh, RANS (realizable k- turbulence closure model) simulations were also investigated to ensure correct results for the spinning wall boundary condition simulations. For these time-accurate simulations, a stationary wall boundary condition was imposed on the body. The spinning motion was imparted on the body by moving the entire mesh at the desired spin rate. An adequate time-step for these simulations was determined to be such that the mesh moved only 0.25° per time step, with 10 sub-iterations per time step.

Implementation of the LNS turbulence model also required a time-accurate solution. The isothermal, rotating wall boundary condition was imposed for the spinning motion of the projectile for simplicity, rather than moving the entire mesh. The LNS calculations were performed at four Mach numbers between Mach 0.70 and 2.7. The time steps used were 3s (Mach 0.70), 2.15 s (Mach 0.94), 1.5 s (Mach 1.25) and 0.8 s (Mach 2.7). These values were determined base on having 70 to 80 time steps within the period of oscillation in the wake flow, assuming a Strouhal number of 0.25. Five inner iterations were performed at each time step. The number of processors used for


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the LNS simulations was increased to 96 such that approximately 78,000 cells were partitioned on each processor. The unsteady calculation took about 30 seconds per time step (5 sub-iterations). The steady-state solutions using the cubic k- turbulence model were used as the starting point for the LNS solutions. The unsteady RANS/LES simulations have currently been run for a total non-dimensional time (based on Strouhal number) of 75. The coefficients presented from this study are the average coefficients over the last 50 non-dimensional times.

2. Pitch Damping Calculation – Lunar Coning The pitch damping coefficient sum calculation follows the procedures described by Weinacht et al.11

and DeSpirito et al.12 for axisymmetric bodies. Their approach uses a specific combination of steady-state spinning and coning motions that allows the pitch damping force and moment coefficient to be directly related to the aerodynamic side force and moment within given constraints. The constraints are that both the coning rate and the angle of attack must be small. The flow may become unsteady at high coning rates or high angles of attack in much the same way the flow over a body at fixed angle of attack at high incidence can become unsteady due to vortex shedding. The use of the combined spinning and coning motion in Ref. 11 was an improvement over previous techniques that simply used a lunar coning motion (no prescribed rotation about the projectile axis) for prediction of the pitch damping coefficients. In both cases, the projectile axis is oriented at a constant angle with respect to the free stream velocity vector. The steady-state, rotating reference frame methodology implemented in CFD++ limits the analysis to lunar coning, since a separate spinning boundary condition about an arbitrary axis could not be appropriately used in conjunction with the rotating reference frame. Therefore, the following discussion will be limited to lunar coning. The results of this methodology have previously been validated using CFD++ in Ref. 12.

The aerodynamic side force and moment coefficients acting on a projectile in steady coning motion can be related to the pitch damping force and moment coefficients. In steady, lunar coning motion, the angular velocity of the projectile results purely from the rotation, Ω, of the projectile about the free stream velocity vector. The angular velocity includes a component along the projectile’s longitudinal axis, which by definition is the spin rate of the

projectile in the non-rolling coordinate system,11 tp cos , where 22 t . The motion can be

decomposed into a combination of two orthogonal planar pitching motions; there is no rotation of the pitch plane with respect to the body. The boundary conditions in the coning frame do not introduce any time dependency into the problem when observed from the coning reference frame, so the resulting flow field is expected to be steady for small angles of attack and small coning rates. Therefore, only steady-state computations are required.

The detailed relationship between the side moment due to coning motion and the pitch damping moment coefficient was developed in Ref. 11 is summarized here. Note that the dynamic coefficients were nondimensionalized by Vd in Ref. 11, while here, as in Ref. 12, the convention of Vd 2 is used. The moment

formulation cast in terms of the in-plane and side moments is written as follows:

mmnmnm CC

V

dC

V

diCiCC

qp 22

(1)

The in-plane moment (real part) results only from the pitching moment, while the total side moment (complex part) consists of contributions from the Magnus moment and the pitch damping moment. The side moment can be written as

mmnn CCC

V

dC

qp

2

(2)

with the Magnus component due to the component of angular velocity along the longitudinal axis of the projectile. The right-hand side of Eq. (2) is simply the variation of side moment with coning rate,

mmn

nn CCC

Vd

CC

qp

2

(3)


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For bodies of revolution, such as the 0.50-cal projectile, a single computation at a non-zero coning rate is sufficient because the side moment for a projectile with a zero coning rate is zero.

3. Pitch Damping Calculation – Transient Planar Motion A variety of approaches exist for processing the computed time-dependent force and moment data to extract the

pitch damping force and moment from the planar pitching motion.12,17,18,19 In the current effort, the total normal force and pitching moment are assumed to have the functional dependence consistent with a linear force and moment expansion. When simplified for the case of planar pitching motion, the total pitching moment has the form shown in Eq. (4). A similar expansion exists for the force coefficient.

V

dqCCCC o

mmmeanmm q 2 (4)

meanmC is obtained directly from the static calculation at the mean angle of attack, mean, and qo is the pitch rate that

can be related to the reduced frequency, k, by

2

(5)

where o is the amplitude of the pitching motion. Based on the findings of Ref. 7 – 11, a reduced frequency of 0.1 was found to be adequate for the current study. The angle of attack and angular rate are determined from the forced sinusoidal variation shown in Eq. (6). sin

(6)

In the current study an mean of 0° and an o of 0.25° were chosen as the pitch-damping moment is known not to vary significantly at small angles of attack. Additionally, is the angle of attack rate when the maximum amplitude is traversed. Using differentiation, it can be also be shown that

o

oo

q

(7)

Time-accurate simulations with a time period, T, equal to are performed using 200 time steps per period.

For the results presented here, the predicted total normal force and pitching moment were well-represented by the assumed form once the initial transients from the initiation of the planar pitching motion damped out. At two times during each cycle a local extrema occurs where mean (i.e. 180,0to ). Cm at 0to is used in equation (4)

from which the pitch-damping moment sum, mm CC

q , can be easily determined. The same procedure can be

used to extract the normal force and pitch damping force coefficients.

III. Results and Discussion The study investigated a Mach number range from 0.6 to 2.7.

Angles of attack from 0° to 9° were investigated during the course of the study. Simulations were not completed for every Mach number and angle of attack on each mesh. However, enough simulations were completed to be able to reasonably determine trends (subsonic, transonic, supersonic). Free stream conditions were chosen to be as close to range conditions as possible: static pressure was set at 101325 N/m2 and static temperature was set at 292 K. Table 1 lists the Mach numbers and corresponding spin rates.

Current CFD results are compared to results from experimental range tests3,4,5 and previous CFD results.7 Some of current results have been previously publish in abridged form12, but are included here for completeness. The static coefficients and the dynamic stability derivatives are discussed separately.

Table 1. Mach numbers, velocities and roll rates utilized in CFD simulations.

Mach No.

Velocity (m/s)

Spin Rate (rad/sec)

0.60 205.52 3389.2 0.70 239.77 3927.9 0.85 291.15 4768.5 0.90 308.28 5050.3 0.94 321.98 5274.7 0.98 335.68 5499.2 1.05 359.68 5892.0 1.10 376.78 6172.5 1.25 428.16 7014.2 1.50 513.79 8417.0 2.00 685.06 11222.8 2.70 924.83 15150.6


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Figure 5. Zero-yaw drag coefficient turbulence model and meshcomparison for ICEMCFD meshes.

A. Static Coefficients The previous RANS study completed by the author7 produced reasonable comparison with range test data that

was available at the time.3, 4 However, since multiple new meshes have been created, a much newer version of CFD++ was utilized, and additional turbulence models were investigated, a comparison of the static aerodynamic coefficients for validation purposes was again conducted.

1. Axial (Drag) Coefficient The zero-yaw drag,

0DC , is compared first with remarkably good agreement to experimental data3 when the

cubic k- turbulence model is utilized for turbulence closure on either 160 cell circumferential ICEMCFD mesh using CFD++v7.1.1 (Figure 5). The two-equation realizable k- turbulence model does not do quite as well, but does significantly better than when it was used in CFD++v3.1,7 which can definitely be attributed to improvements in the code between versions. However, the three-equation k-R over predicts , especially in the transonic regime on the one mesh that it was investigated. The boundary layer is obviously not being well predicted by this model. On the GRIDGEN mesh (results not shown), both the cubic k- model and the three-equation k-R solution were implemented with no significant differences noted with the ICEMCFD meshes. Additionally, no significant changes occurred in using the realizable k-turbulence closure model when CFD++ v10.1 used, circumferential resolution or boundary layer resolution was increased, or when a rotating mesh was used to implement spin, rather than a boundary condition. This demonstrates grid independence for this coefficient, for RANS turbulence models. Most of the LNS solutions agree reasonably well with the comparable steady-state, cubic k- model solution. However, there is a significant increase for the time-accurate LNS model at Mach 0.94. The cause of the disagreement at Mach 0.94 is unclear. Perhaps the solution has not yet converged.

A “best model” will be selected for this geometry type (boat tail, fillet, flat base) once all static coefficients have been compared. The use of the cubic k- turbulence model would be the most efficient if LNS simulations were going to be completed, as then only one set of steady-state calculations would need to be completed as this is the turbulence model that is used to initialize the LNS calculations.

Both the cubic k- turbulence model and the realizable k- turbulence model show good agreement for axial force coefficient, CX, at angle of attack. Grid convergence and version independence was established at = 2° (Figure 6) using the realizable k-turbulence closure model. The 160 circumferential cell mesh is the RANS II mesh.


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Figure 6. Drag coefficient at a = 2°, showing mesh independence.

Figure 7. Normal force coefficient derivative at = 2° for 160circumferential cell ICEMCFD mesh.

2. Normal Force Coefficient Derivative The normal force coefficient,

NC , is well predicted except near Mach 1 for all meshes and turbulence models.

Figure 7 shows only the v7.1 results of the160 circumferential ICEMCFD meshes. The remainder of the meshes and simulation strategies show the same trends. The scatter in the experimental data near Mach 1 makes it difficult to determine which turbulence model is the best to use, although the realizable k-turbulence closure model may be slightly better. Above Mach 1, all the computational results agree quite well. It is interesting to note the DREV5 experimental results at Mach 1.5 and the M334 results at Mach 2 and Mach 2.5 are noticeably larger than either the other range results or the computed CFD results, which may indicate additional experimental error.


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Figure 8. Pitching moment coefficient derivative at = 2° for 160circumferential cell ICEMCFD meshes.

3. Pitching Moment Coefficient

The pitching moment coefficient, mC , shows very good agreement with the experimental data, regardless of

the turbulence closure model or mesh (Figure 8) utilized. The experimental data also shows good agreement between tests. Therefore, based on static coefficients only, either the realizable k- or the cubic k- turbulence closure model could be used for this type of geometry (i.e. long boattail, fillet, flat base).

B. Dynamic Derivatives

1. Roll Damping Coefficient The roll damping coefficient,

plC , is typically well predicted by the steady-state RANS simulation at = 0°. A

steady-state simulation with a rotating boundary condition can be used as this is an axisymmetric spinning projectile. The turbulence closure model chosen will likely affect the results as the viscous forces (not just the pressure forces) are a large contributor to this coefficient.

Comparison of the CFD shows that the turbulence model does affectpl

C slightly (Figure 9). The largest affect

seems to be a combination of grid and turbulence model at subsonic Mach numbers. At Mach 0.6 and Mach 0.7, the three equation k-e-R model on the Gridgen mesh agrees with the other data. However, implementing the same turbulence model on the RANS I mesh produce significantly more roll damping at these Mach numbers. As this is the only coefficient that shows a discrepancy, it is not of great concern to the author as the three equation k-e-R model will not likely be implemented in future studies. Agreement between the RANS model using the cubic k- turbulence model and the LNS solution (the RANS portion uses the cubic k turbulence model) shows that a steady-state solution is able to adequately determine

plC . The time-accurate, moving-mesh, RANS simulations (data not

shown) also agreed with the steady-state solution. The experimental M8 and M33 data from Ref. 4 is not shown here as it was recently discovered that the

plC used

in the range reduction was obtained from an empirical fit, as spin data was not available from the range tests. This explains the large discrepancies in

plC that were reported in Ref. 7. With the addition of the Canadian (DREV)

data5, two experimental data points were now available for comparison. Additional experimental data, especially at subsonic Mach numbers, would be useful for validation purposes.


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Figure 9. Roll damping moment coefficient

2. Magnus Moment Coefficient Reference 7 found that the realizable k- turbulence model accurately predicted the Magnus moment coefficient,

pnC , at = 5°, but not at = 2°. Agreement with experimental data remains good at = 5°, so presentation of

data in this paper is limited to = 2°. Additional work at ARL9, 10, 12 has found that the use of LNS may improve agreement with experimental data.

Therefore, it was decided that in addition to investigating additional steady-state RANS turbulence models, a time accurate LNS solution would also be obtained. To achieve this, multiple meshes were created. The initial mesh, RANS mesh I, was created with the LNS solution in mind, but was not fully isotropic in the wake region in an attempt to reduce the mesh size. Further steady-state simulations with different turbulence closure models did not show improvement in the transonic and subsonic regimes over those in Ref. 7. However, there were improvements in the supersonic regime indicating possible improvements in the newer version of the solver (Figure 10) shows that while varying the turbulence closure model for the RANS simulations did affect the magnitude of a little, there

was no improvement in the CFD following the experimental data trend. While Ref. 7 proposed that the discrepancy may be due to the lack of engraving being modeled on the projectile, Weinacht20 and Silton21 have shown that the lack of projectile engraving does not significantly effect . Therefore, an LNS solution was also obtained on the

mesh. A limited Mach number range was investigated for the LNS simulations in order to limit the computational time

required. The set of Mach numbers investigated (Mach 0.7, 0.94, 1.25, 2.7) was chosen to test each flow regime. Figure 10 shows that for the limited data available, the LNS simulations on the hexahedral mesh are beginning to predict the data trends, except at Mach 0.94, where the coefficient derivative remains positive, although it did decrease over the steady-state cubic k- turbulence closure model. As the LNS solution appeared promising on this mesh, the author decided it was a good use of resources to create a more uniform wake region for a better LNS mesh, with the thought that more improved results would be obtained.

During the creation of the LNS mesh, the RANS II mesh was also created (the wake region of the RANS mesh was replaced with an LES mesh). Therefore, steady-state simulations were completed on the RANS II mesh. Steady-state simulations were also completed on the LNS mesh using the cubic k- model as it was the required starting point for the time-accurate LNS simulations. For both steady-state simulations, a slight improvement in magnitude was observed in the subsonic regime, but otherwise the results remained the same (Figure 11). Again, for the LNS simulations only a subset of Mach numbers were completed (Mach 0.94, 1.25, 1.5, 2.7). Results above Mach 1 were similar to the previous LNS simulations (Figure 10 and Figure 11). However, the results actually showed larger discrepancies with experimental data at Mach 0.94 – there was no difference between the steady-state


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Figure 11. Magnus moment coefficient derivative at = 2°: LNS mesh.

Figure 10. Magnus moment coefficient derivative at = 2°: initial study.

simulation and the time-accurate LNS simulation. Additional transonic and subsonic Mach numbers (especially Mach 0.7) could prove useful in determining the effectiveness of the LNS simulations.

A switch to CFD++ v10.1 was made as the older version was no longer supported on DSRC computer system. A subset of Mach numbers was rerun in the new version, with no significant differences observed. Other discussions taking place at this time also indicated that increased circumferential resolution could improve the steady-state simulation results, as could a smaller growth rate in the boundary layer. The circumferential resolution was doubled with no noticeable difference at the Mach numbers investigated (Mach 0.7, 0.94, 0.98, 1.05, 1.25, 1.5, 2.7). The same held true for the smaller boundary layer growth rate. The effect of using a time-accurate, moving mesh RANS calculation rather than a steady-state boundary condition on was also investigated at Mach 0.7, Mach 0.98, and


13

Figure 12. as a function of angle of attack at Mach 0.7, Mach 0.98 and Mach 1.5.

Mach 1.5. There were no significant differences for Mach 0.98 or Mach 1.5. Differences were observed in the side moment (Cn) at Mach 0.7, but the values were so near zero, that they were deemed insignificant.

The values of Cn determined from the CFD simulations are quite small (on the order of 0.001) and could be contributing to the discrepancies between the CFD and experiments. One way to assess the accuracy of the CFD is to evaluate a more complete range of angles of attack for a given Mach number and then complete a curve fit to the data. Often Magnus moment coefficient has a bi-modal behavior – cubic with respect to sin at small angles of attack and linear at larger angles of attack. Three Mach numbers were investigated for this behavior (Mach 0.7, 0.98, 1.5) at seven angles of attack (0°, 1°, 2°, 3°, 5°, 7°, 9°). Figure 12 shows the results. The bi-modal behavior clearly exists at each Mach number. At Mach 0.98, the cubic behavior is troubling. The cubic term would normally be a large positive number. At Mach 0.98, a good cubic fit requires that the cubic term be negative. While this is not a clear indication of overall behavior, it does indicate that the physics near Mach 1 may not be correct. Further investigation into sources of error needs to be completed.

It is imperative that the Magnus moment can be properly predicted and understood. For spin-stabilized projectiles, it is one of the main causes of dynamic instability leading to trim angles down range. Until the trends in the non-linear regime can be accurately predicted, the understanding of the causes and how to mitigate them will continue to elude researchers.

3. Pitch-damping Moment Coefficient The final coefficient necessary to complete the suite of dynamic stability derivatives is the pitch-damping

moment coefficient sum, mm CC

q . The CFD results presented here were completed using the lunar coning

methodology at = 2° for the realizable k- turbulence model on the ICEMCFD RANS II mesh as well as the k-R turbulence model on the GRIDGEN mesh. The results of calculating the pitch-damping moment coefficient using the planar pitching methodology (mean = 0°) have also been obtained for the realizable k- turbulence model on the ICEMCFD RANS II mesh over a more limited Mach number range. Figure 13 shows very good agreement between the methods, meshes, and turbulence models, except at Mach 1.05 and Mach 1.1, where the computational lunar coning results differ by 20% to 50%. The cause of this discrepancy in unclear. Also, the decrease in the pitch


14

Figure 13. Pitch-damping moment coefficient sum.

damping in this same Mach number has been previously reported by DeSpirito et. al,12 although the cause remains a mystery at this time. The current experimental data indicates that it might be a real flow feature. Agreement with the experimental data above Mach 2 is very good. At Mach 1.5, the computations predict slightly more pitch damping that is seen in the range tests. In the transonic and subsonic regimes, the agreement with the experimental data may not be as good. However, the scatter in the experimental results makes it difficult to make any type of definitive statement. The positive values of

mm CCq originally obtain by McCoy3 are not realistic and possibly

indicate an interaction with Magnus moment coefficient that is not being properly accounted for in the data reduction. The near zero value determined by ARFDAS4 (Exp – Single Fit, Exp – Group Fit) just below Mach 1 is more probable. The computational results appear to “split” the difference in the scatter of the experimental data, further indicating that the experimental results are not properly separating the Magnus moment coefficient from the pitch-damping moment coefficient sum.

C. Projectile Stability The most important part of any study to obtain the aerodynamic coefficients and dynamic derivatives, whether

through computations of experiments, is whether the correct projectile behavior, as seen in flight, can be predicted. This can be achieved through a linear theory stability analysis or complete trajectory simulations. A linear theory stability analysis was completed here in order to better establish if the differences being observed between the computational and experimental Magnus moment coefficient at = 2° in the subsonic and transonic regime are in part due to the experimental coupling with pitch-damping moment coefficient. A complete derivation of the linear theory stability analysis can be found in McCoy22 in the old BRL (predecessor of ARL) notation. This has been converted to the newer convention as used in ARFDAS and PRODAS.4 It is the newer format that is shown below (Equations 1 – 5).

The main portion of the stability analysis for a spinning projectile, assuming it is gyroscopically stable (which this one is), is determination of the fast and slow mode damping exponents. The fast and slow mode damping exponents (F,S) are determined by

, 21

12

(8)


15

where

4 (9)

(10)

(11)

1

(12)

11

(13)

and 2

(14)

is the gyroscopic stability factor, is the free stream air density (1.209 kg/m3), m is the projectile mass (0.4198

kg), Ix is the axial moment of inertia (7.84e-7 kg-m2), and Iy is the transverse moment of inertia (7.39e-6 kg-m2). Typically the frequencies are plotted versus angle of attack for a given Mach number in order to determine what

angle, if any a limit cycle may occur. However, for the analysis completed here, the frequencies are plotted for a given angle of attack versus Mach number for a direct comparison to the experimental data (Figure 14). Almost all the CFD data is taken from the 240 circumferential cell ICEMCFD mesh, except for . That coefficient is

taken from the lunar coning calculations completed on the ICEMCFD RANS II mesh. At = 5°, the agreement between computational and experimental calculations is quite good for the fast mode. For the slow mode, the agreement between the experimental data and the computational results is not that good, although stability is still predicted for by the CFD. All trends are as expected for both the fast and slow mode frequencies across the Mach number range. The exact values of a given mode (i.e. slow or fast) differ by nearly 50% between the computation results and the experimental results. However the numbers are quite small to begin with, so the differences are in the thousandths place, which is quite good. Unfortunately, this agreement once again does not occur for = 2°, indicating that it is not just the coupling between Magnus moment and pitch-damping moment coefficients causing the discrepancies. The CFD compares well for the two highest Mach numbers, but begins diverging from the experimental results by Mach 1.25. The trend in the transonic regime is completely wrong, and can be attributed to the discrepancies in the Magnus moment coefficient.

IV. Conclusions The prediction of dynamic stability derivatives, roll damping, Magnus, and pitch damping moments, along with

the static aerodynamic coefficients were presented for the 0.50-cal spin-stabilized projectile over a range of Mach numbers. The static aerodynamic coefficients were generally well predicted via steady-state CFD methods across the flight regimes (subsonic to supersonic). Both the cubic k-turbulence model and the realizable k-turbulence model typically did a good job predicting the coefficients, much better than the three equation k-R turbulence model. The roll damping moment was also shown to be adequately predicted via steady-state CFD methods when compared to the experimentally determined roll data. For the Mach numbers investigated to date, the pitch damping moment coefficient is well predicted using either the steady-state lunar coning methodology or the transient planar pitching methodology, regardless of mesh or turbulence model utilized. There is some discrepancy at transonic Mach numbers, which has been seen before, but cannot be accounted for at this time. Finally, for the Mach numbers, meshes and turbulence models investigated here, the non-linearity in Magnus moment cannot be correctly predicted at small angles of attack. Although the typical bi-modal behavior is being seen at the Mach numbers investigated, the values and trends across Mach numbers don’t agree with experimental data. It is still unclear if the use of time-accurate LNS simulation will improve the CFD predictions of the Magnus moment coefficient, though


16

Figure 14. Fast and slow mode frequency comparison for = 2° (top) and = 5° (bottom).

indications are that it may to some extent. An initial look at the projectile stability shows that these discrepancies in Magnus moment will also affect the simulated flight of the projectile, which is unacceptable as a final result. Therefore, the Magnus moment portion of this study will be continued (and extended to other configurations) until the cause of the discrepancy can be isolated and corrected. In future studies, additional meshing and simulation strategies will be undertaken. They include: the effect of further decreasing the wall boundary spacing (y+ near 0.1), a more in-depth time step study for LNS, additional Mach numbers for the LNS simulations, the use of the virtual range technique23 (RANS and LNS turbulence models), and a more refined LNS mesh.

Acknowledgments The author thanks Dr. Paul Weinacht and Dr. James DeSpirito, Army Research Laboratory, and Wayne

Hathaway, ArrowTech Associates, for many helpful technical discussions. This work was supported in part by a grant of high-performance computing time from the U.S. Department of Defense High Performance Computing Modernization program at the ARL Distributed Shared Resource Center, Aberdeen Proving Ground, Maryland.


17

References

1 Weinacht, P., “Projectile Performance, Stability, and Free-Flight Motion Prediction Using Computational Fluid Dynamics,” AIAA J. Spacecraft & Rockets, Vol. 4, No. 2, 2004, pp. 257–263. 2 Hitchcock, H.P., “Aerodynamic Data for Spinning Projectiles,” BRL Report No. 620, October 1947. 3 McCoy, Robert L., “The Aerodynamic Characteristics of .50 Ball, M33, API, M8, and APIT, M20 Ammunition,” BRL-MR-3810, January 1990. 4 Whyte, B. Personal communication. Arrow Tech Corporation, August 2002. 5 Dupuis, A.D., Bernier, A. and Hathaway, W., "Data Compendium of the free-flight aerodynamic characteristics of 0.50 Cal Configurations for Range Limited Application at Supersonic Speeds" DREV TR 1999-060, November 1999. 6 Guidos, B.J. and Chung, S.K., “Computational Flight Design of .50 Caliber Limited Range Training Ammunition,” ARL-TR-662, January 1995. 7 Silton, S. I., “Navier-Stokes Computations for a Spinning Projectile from Subsonic to Supersonic Speeds,” AIAA J. of Spacecraft & Rockets, Vol. 42, No. 2, 2005, pp. 223–231. 8 DeSpirito, J., and Heavey, K. R., “CFD Computation of Magnus Moment and Roll Damping Moment of a Spinning Projectile,” AIAA-2004-4713, Aug. 2004. 9 DeSpirito, J., and Plostins, P., “CFD Prediction of M910 Projectile Aerodynamics: Unsteady Wake Effect on Magnus Moment,” AIAA-2007-6580, Aug. 2007. 10 DeSpirito, J., “CFD Prediction of Magnus Effect in Subsonic to Supersonic Flight,” AIAA-2008-0427, Jan. 2008. 11 Weinacht, P., Sturek, W. B., and Schiff, L. B., "Navier-Stokes Predictions of Pitch Damping for Axisymmetric Projectiles,” AIAA J. Spacecraft & Rockets, Vol. 34, No. 6, 1997, pp. 753–761. 12 DeSpirito, J, Silton, S., and Weinacht, P., “Navier-Stokes Predictions of Dynamic Stability Derivatives: Evaluation of Steady-State Methods,” J. of Spacecraft and Rockets, Vol. 46, No. 6, Nov.2009. 13 SolidWorks, Inc., “SolidWorks User’s Manual,” Concord, MA, 2007. 14 Ansys, Inc., “ICEM CFD User’s Manual,” Canonsburg, PA, 2007. 15 “GRIDGEN Users Manual,” Pointwise, Inc., Bedford, TX, 2006. 16 Metacomp Technologies, Inc., “CFD++ User’s Manual,” Agoura Hills, CA 2007. 17 Park, S.H. and Kwon, J.H., “Navier-Stokes Computations of Stability Derivatives for Symmetric Porjectiles,” AIAA 2004-0014, Jan. 2004 18 Oktay, E. and Akay, H.U., “CFD Predictions of Dynamic Derivatives for Missiles,” AIAA 2002-0276, Jan. 2002. 19 Stalnaker, J.F. and Robinson, M.A., “Computations of Stability Derivatives of Spinning Missiles Usin Unstructured Cartesion Meshes,” AIAA 2002-2802, June 2002. 20 Weinacht, P., “Validation and Prediction of the Effect of Rifling Grooves on Small-Caliber Ammunition Performance,” AIAA-2006-6010, August 2006. 21 Silton, S. and Webb, D., “Experimental Determination of the Effect of Rifling Grooves on the Aerodynamics of Small Caliber Projectiles,” AIAA-2006-6009, August 2006. 22 McCoy, Robert L., Modern Exterior Ballistics, Schiffer Military History, Atglen, PA, 1999, pp. 232-233. 23 Sahu, J., “Unsteady Free-Flight Aerodynamics of a Spinning Projectile at a High Transonic Speed,” AIAA-2008-7115, August, 2008.

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