ARL-SR-0337 ● SEP 2015

US Army Research Laboratory

Flight Behaviors of a Complex Projectile Using 

a Coupled Computational Fluid Dynamics 

(CFD)-based Simulation Technique: Free 

Motion

by Jubaraj Sahu and Frank Fresconi

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ARL-SR-0337 ● SEP 2015

US Army Research Laboratory

Flight Behaviors of a Complex Projectile Using 

a Coupled Computational Fluid Dynamics 

(CFD)-based Simulation Technique: Free 

Motion

by Jubaraj Sahu and Frank Fresconi

Weapons and Materials Research Directorate, ARL

Approved for public release; distribution is unlimited.
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Flight Behaviors of a Complex Projectile Using a Coupled Computational Fluid 

Dynamics (CFD)-based Simulation Technique: Free Motion

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Jubaraj Sahu and Frank Fresconi

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14. ABSTRACT

This report describes a computational study to understand the free roll, pitch, and yaw motion of a canard-controlled, finstabilized projectile. Numerical computations were performed for this projectile using an advanced coupled computational 

fluid dynamics and rigid body dynamics technique. Additionally, the coupled approach was tailored to be constraints-based 

for the rigid body dynamics allowing for pure pitching and pure rolling motions to be studied prior to considering the full rollpitch-yaw motion. Flow visualizations and aerodynamic force and moment computations indicate unsteady canard stall at high 

angle of attack and significant interaction of these canard vortices on the afterbody-fins. Parameter estimation was performed 

to obtain aerodynamic coefficients, assess the aerodynamic model, and understand the roll-pitch-yaw coupling. These results 

suggest that improvements to aerodynamic modeling are necessary to better capture high angle-of-attack phenomenon such as 

unsteady canard stall and vortex interactions. Analysis of the pure pitch and pure roll predicts the coupled roll-pitch-yaw 

motion well with the exception of unsteady and interaction effects during periods of high angle of attack. 

15. SUBJECT TERMS

coupled method, virtual fly-out, pitch, roll-pitch-yaw motion, parameter estimation

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iii

Contents

List of Figures iv

List of Tables v

Preface vi

Acknowledgments vii

1. Introduction 1

2. Computational Methodology 3

3. Coupled CFD/RBD Approach 4

4. Aerodynamics Modeling 5

5. Model Geometry and Grid 9

6. Results 11

6.1 Pure Pitching Case 17

6.2 Pure Rolling Case 24

6.3 Roll-Pitch-Yaw Case 26

7. Conclusions 31

8. References 33

Appendix. Aerodynamic Coefficients at Mach 0.6 37

List of Symbols, Abbreviations, Acronyms, and Nomenclature 39

Distribution List 41
iv

List of Figures

Fig. 1 Body-fixed coordinate system and aerodynamic angles........................5

Fig. 2 Computational model of the projectile geometry ..................................9

Fig. 3 Expanded view of the mesh in the XZ plane near the body ................10

Fig. 4 View of the mesh in the circumferential plane, near the canard before 

overset..................................................................................................10

Fig. 5 View of the mesh in the circumferential plane, near the canards after 

overset cut ............................................................................................11

Fig. 6 Particle traces at 3 different instants in time during roll-pitch-yaw 

motion, M = 0.6, (a) t = 0.03, (b) t = 0.15, and (c) t = 0.26 s ..............13

Fig. 7 Vorticity contours at different instants in time during initial phase of 

roll-pitch-yaw motion, M = 0.6, 0.02 ≤ t ≤ 0.07 s ...............................14

Fig. 8 Surface pressure contours at different instants in time during initial 

phase of roll-pitch-yaw motion, M = 0.6, 0.02 ≤ t ≤ 0.07 s.................16

Fig. 9 Expanded view of surface pressure contours near the fins at different 

instants in time during initial phase of roll-pitch-yaw motion, M = 0.6, 

0.02 ≤ t ≤ 0.07 s ...................................................................................17

Fig. 10 Euler pitch angle (left) and pitch rate (right) for pure pitching case ...17

Fig. 11 Axial force coefficient of body-fin (top left), normal force coefficient 

of body-fin (top right), roll moment coefficient of body-fin (middle 

left), pitching moment coefficient of body-fin (middle right), and side 

moment of body-fin (bottom) from pure pitching motion...................19

Fig. 12 Axial force coefficient of canard (top left), normal force coefficient of 

canard (top right), roll moment coefficient of canard (bottom left), and 

pitching moment coefficient of canard (bottom right) from pure 

pitching ................................................................................................20

Fig. 13 States from parameter estimation analysis of pure pitching (Euler pitch 

angle root-sum-square fit error 0.15°) .................................................22

Fig. 14 Pitching moments from parameter estimation analysis for pure 

pitching (top left = body-fin, top right = individual canard, bottom = 

total).....................................................................................................23

Fig. 15 States from parameter estimation analysis of pure rolling (Euler roll 

angle root-sum-square fit error 0.1°) ...................................................24

Fig. 16 Rolling moments from parameter estimation analysis for pure rolling 

(top left = body-fin, top right = individual canard, bottom = total).....25

Fig. 17 Axial force coefficient of body-fin (top left), normal force coefficient 

of body-fin (top right), roll moment coefficient of body-fin (bottom 

left), and pitching moment coefficient of body-fin (bottom right) from 

roll-pitch-yaw motion ..........................................................................26
v

Fig. 18 Axial force coefficient of canard (top left), normal force coefficient of 

canard (top right), roll moment coefficient of canard (bottom left), and 

pitching moment coefficient of canard (bottom right) from roll-pitchyaw motion...........................................................................................27

Fig. 19 Euler roll (top left), pitch (top right), and yaw (bottom) angles from 

parameter estimation analysis of roll-pitch-yaw (root-sum-square fit 

errors of 0.32°, 0.19°, and 0.19° in Euler roll, pitch, and yaw angles)28

Fig. 20 Roll (top left), pitch (top right), and yaw (bottom) rates from 

parameter estimation analysis of roll-pitch-yaw..................................29

Fig. 21 Total moments from parameter estimation analysis for roll-pitch-yaw 

(top left = roll, top right = pitch, bottom = yaw) .................................30

Fig. 22 Rolling moment coefficient of body-fin (top left), pitching moment 

coefficient of body-fin (top right), side moment coefficient of body-fin 

(middle), rolling moment coefficient of canard (bottom left), and 

pitching moment coefficient of canard (bottom right) from direct 

method and parameter estimation of roll-pitch-yaw............................31

List of Tables

Table 1 Initial conditions for coupled simulations............................................12

Table 2 Aerodynamic coefficients for body-fin at Mach 0.6 from direct 

analysis of pure pitching ......................................................................21

Table 3 Aerodynamic coefficients for canard at Mach 0.6 from direct analysis 

of pure pitching....................................................................................21

Table 4 Aerodynamic coefficients at Mach 0.6 from parameter estimation of 

pure pitching, m = –80.7 C q .....................................................................24

Table 5 Aerodynamic coefficients at Mach 0.6 from parameter estimation of 

pure rolling, l = –12.1 C p .........................................................................25

Table A-1 Aerodynamic coefficients at Mach 0.6 from direct and parameter 

estimation analysis of roll-pitch-yaw...................................................38
vi

Preface

The paper “Flight Behaviors of a Complex Projectile using a Coupled CFD-based 

Simulation Technique: Free Motion” was originally published in Proceedings of 

the 33rd AIAA Applied Aerodynamics Conference (2015 June 22–26, Dallas, TX). 

This version contains a few important minor modifications of the original 

publication based on additional reviews. 
vii

Acknowledgments

This work was accomplished as part of a grand challenge project jointly sponsored 

by the Department of Defense High Performance Computing Modernization 

program and the US Army Research Laboratory. The author also wishes to thank 

Dr Sidra Silton and Ms Karen R Heavey for their help with the computational 

meshes, Dr Sukumar Chakravarthy of Metacomp Technologies for providing 

expert advice on the coupled simulations, and Mr Richard Angelini for his help 

with advanced flow visualizations.
viii

INTENTIONALLY LEFT BLANK.
1

1. Introduction

Accurate determination of flight behaviors is critical to the development of new, 

affordable munitions. Various techniques such as the semiempirical,1,2 wind 

tunnel,3,4 free-flight,5–7 and computational fluid dynamics (CFD).

8–10 are used 

routinely for aerodynamic characterization of projectiles. Rigid body flight 

dynamics often utilize aerodynamic coefficients obtained using these techniques 

within an aerodynamic model framework. A major concern, especially when 

adding moveable aerodynamic surfaces (MAS) (e.g., canards) upstream of the body 

or stabilizing fins, is flow interactions.11,12 The motion of upstream control surfaces 

at various aerodynamic angles of attack and Mach numbers greatly influences the 

pressure distribution on downstream surfaces. Even for unguided flights, these flow 

interactions exist and must be accurately taken into account in the design analysis. 

A significant contribution to these studies would be shifting from multidimensional 

table look-ups based on exhaustive aerodynamic quantification to mathematical 

models that inherently include effects captured in the multidimensional tables.

Fortunately, improved computer technology and state-of-the-art numerical 

procedures have continued to enable solutions to complex, 3-dimensional (3-D) 

problems associated with projectile and missile aerodynamics.8–10,12–14 In 

particular, CFD has become a powerful tool for the prediction of aerodynamics and 

for subsequent design and performance evaluation of these weapon systems. It is 

essential that CFD be brought into the earlier stages of design and development to 

be more effective. Our recent focus has therefore been directed at the development 

and application of advanced predictive capabilities to compute the unsteady 

aerodynamics of both unguided and guided projectiles at all speeds from subsonic 

to supersonic speeds. Accurate numerical modeling of the unsteady aerodynamics 

has been found to be challenging and has required the use of time-accurate solutions 

techniques.

New advanced computational techniques are being developed to understand flight 

behaviors of both unguided and guided projectiles. One such technique involves 

coupling of CFD and rigid body dynamics (RBD) codes for the simulation of 

projectile free flight motion in a time-accurate manner. This technique, known as 

coupled CFD/RBD or virtual fly-out method,15 uses advanced CFD methods to 

characterize the unsteady aerodynamics at each instant in time during flight. In the 

coupled CFD/RBD procedure, the aerodynamic forces and moments are computed 

at every time step in the CFD solver and transferred to the RBD code. The RBD 

code computes the projectile body’s response to the forces and moments. The
2

output of RBD state variables are transferred back to the CFD flow solver, which 

then computes the aerodynamic forces and moments at the next time step subject 

to the updated RBD states. This process marches forward in time from a given set 

of initial conditions for the fly-out simulations. This coupled technique provides 

both the unsteady aerodynamics and the flight dynamics in an integrated manner. 

It yields a wealth of data unavailable in experimental methods, but it does involve 

highly computer-intensive calculations requiring large computational resources. 

Flow fields, pressure distributions (and hence forces and moments) on various 

surfaces, and the complete 12-state RBD history are available from the coupled 

solutions. The coupled CFD/RBD technique has already been demonstrated for 

various finned- and spin-stabilized projectiles16–20 and validated in some of these 

cases. Computed results obtained from the coupled simulations were validated with 

available free-flight test data at subsonic, transonic, and supersonic speeds. This 

technique was further extended for computation of free-flight aerodynamics and 

flight dynamics of a finned projectile with pulse jet control21 and is being extended 

for guided flights.22,23

The approach for coupled CFD/RBD simulations is to capture static and dynamic 

aerodynamic behavior over short time durations with different motions. Coupling 

the flight dynamics with the fluid mechanics helps ensure that the data are collected 

over the appropriate conditions (e.g., angle of attack, angular rate). Ultimately, this 

method obtains aerodynamic data and enables aerodynamic models to be built and 

verified. Performing coupled simulations in this manner allows for screening of 

situations where conventional aerodynamic models based on static wind tunnel or 

CFD techniques break down. These instances are encountered more often as wider 

classes of munitions (small-medium-large caliber) feature control inputs and the 

associated flow complexity such as interactions, unsteadiness, and high angle of 

attack. Thus, a major benefit of these coupled simulations is to mitigate risk of 

unanticipated flight behaviors during guided, free-flight experiments.

Of particular interest are investigations into flight motions such as the pure pitch, 

pure rolling motion, and a complex roll-pitch-yaw motion. Constraints-based 

coupled fly-out simulation approaches have been developed in this research to 

generate aerodynamics for pure pitch, pure roll, and coupled roll-pitch-yaw 

dynamics with free motion (no control input). The novel contributions of this study 

include using coupled techniques on high maneuverability airframes to identify 

high angle-of-attack effects, evaluating aerodynamic modeling performance, and 

assessing the significance of coupling between degrees of freedom (DOF). This

report details the constraints-based coupled CFD/RBD technique, which is applied 

to a highly maneuverable, canard-controlled, fin-stabilized projectile design to 
3

facilitate complete flight analysis. The following sections describe the CFD method 

used, coupled numerical procedure, aerodynamic modeling, and computed results 

obtained for various flight motions. 

2. Computational Methodology

Time-accurate, coupled CFD/RBD computations were performed using NavierStokes techniques for a canard-controlled projectile at Mach, M = 0.6. Both 

unsteady aerodynamics and flight dynamics were computed simultaneously and the 

flight response of the projectile was determined. In all cases, full 3-D solutions were 

obtained and no symmetry was used. 

The complete set of 3-D time-dependent Navier-Stokes equations is solved in a 

time-asymptote manner to obtain converged steady-state solutions. A commercially 

available code, CFD++,

24,25 is used and the 3-D, time-dependent Reynoldsaveraged Navier-Stokes equations are solved using the following finite volume 

method:

[ ] ∫ ∫ ∫ + − ⋅ =

V V

dV dA dV

t

W F G H

∂

∂ (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.

Several techniques such as implicit scheme and relaxation are used to achieve faster 

convergence. Use of an implicit scheme circumvents the stringent stability limits 

suffered by their explicit counterparts, and successive relaxation allows update of 

cells as information becomes available and thus aids convergence. Second-order 

discretization was used for the flow variables and the turbulent viscosity equation. 

The turbulence closure is based on topology-parameter-free formulations. A 

realizable 2-equation k-ε turbulence model26 was used for the computation of 

turbulent flows. These models are ideally suited to unstructured bookkeeping and 

massively parallel processing due to their independence from constraints related to 

the placement of boundaries and/or zonal interfaces. The basic CFD solution 

technique described here is coupled with RBD for simultaneous prediction of both 

uncontrolled and control flights.
4

3. Coupled CFD/RBD Approach

Research efforts are ongoing to perform time-accurate multidisciplinary coupled 

CFD/RBD computations for complex guided projectiles. With or without flight 

control, the coupled CFD/RBD procedure allows 1) “virtual fly-out” of projectiles 

on the supercomputers and 2) the prediction of actual flight paths of a projectile 

and all the associated unsteady free-flight aerodynamics in an integrated manner. 

A time-accurate numerical approach is used in the coupled virtual fly-out 

simulations. This approach requires that the 6DOF body dynamics be computed at 

each repetition of a flow solver. The CFD capability used here solves the same 

Navier-Stokes equations shown in Eq. 1 and incorporates advanced boundary 

conditions and grid motion capabilities. For time-accurate simulations of virtual 

fly-outs that are of interest here, a dual time-stepping procedure was used to achieve 

the desired time accuracy in the time-accurate solutions. The whole grid was moved 

to take into account the motion of the projectile. To account for RBD, the grid point 

velocities were set as if the grid is attached to the rigid body with 6DOF. 

The 6DOF model for the projectile flight is shown in Eq. 2.

�

xẋ

yẏ

zż

� = �

ccθθccψψ ssφφssθθccψψ − ccφφssψψ ccφφssθθccψψ + ssφφssψψ

ccθθssψψ ssφφssθθssψψ + ccφφccψψ ccφφssθθccψψ + ssφφssψψ

−ssθθ ssφφccθθ ccφφccθθ

� �

uu

vv

ww

�

�

φφ̇

θθ̇

ψψ̇

� = �

1 ssφφttθθ ccφφttθθ

0 ccφφ −ssφφ

0 ssφφ⁄ccθθ ccφφ⁄ccθθ

� �

pp

qq

rr

�

�

uu̇

vv̇

wẇ

� = 1

mm �

XX

YY

ZZ

� − �

0 −rr qq

rr 0 −pp

−qq pp 0

��

uu

vv

ww

�

�

pṗ

qq̇

rṙ

� = II

⃑−1 �

LL

MM

NN

� − II

⃑−1 �

0 −rr qq

rr 0 −pp

−qq pp 0

�II

⃑�

pp

qq

rr

�

 (2)

These equations are numerically integrated forward in time to obtain the flight 

dynamic response. The dynamics are constrained by setting certain moments or 

forces to zero depending on the dynamics being isolated, hence the term 

“constraints-based”. As an example, when examining the pitch-only motion at a 

specific Mach number, all forces and the roll and yaw moments are set to zero.

Typically, the coupled solution procedure requires 3 steps. First, we begin with a 

computation performed in the “steady-state mode” with the grid velocities 

prescribed to account only for the translational motion component of the complete 

set of initial conditions. At the second step, we also impose the angular orientations 
5

from the initial conditions and spin rate is added for spinning projectiles. 

Computations are performed with the spin in a time-accurate mode for a desired 

number of spin cycles. Converged solution from this second step provides the initial 

condition for the third step. In the third step, the remaining rotational velocity 

components (pitch and yaw) are added and a completely coupled CFD/RBD 

computation is performed in time-accurate mode. The solution from the third step 

should correspond to the complete set of initial conditions that includes all 

translational and rotational velocity components and accounts for initial position 

and angular orientations. 

4. Aerodynamics Modeling

The aerodynamic model relates the aerodynamic data to the aerodynamic forces 

and moments acting on a projectile in flight. Aerodynamic data, in the form of 

coefficients, reduce the complex fluid mechanics around the projectile to a form 

amenable to analysis. Aerodynamic characterization techniques (semiempirical 

aeroprediction, wind tunnel, CFD, spark range, onboard sensor) yield forces and 

moments in the body frame.

An illustration of the projectile with the body-fixed coordinate system (Fig. 1) 

permits definition of the sense of forces and moments and aerodynamic angles. 

Fig. 1 Body-fixed coordinate system and aerodynamic angles

Aerodynamic angles are computed based on the body-fixed velocity components. 

The pitch angle of attack (or sometimes just angle of attack) is defined as

αα = aa aaaa � ww

√uu2 + vv2 + ww2

� (4)
6

The yaw angle of attack (or sometimes sideslip angle) is

ββ = aa aaaa � vv

√uu2 + vv2 + ww2

� (5)

The aerodynamic roll angle is

φφAA = aa aaaa �

vv

ww� (6)

Finally, the total angle of attack is the root-square-sum of the pitch and yaw angles 

of attack:

αα� = �αα2 + ββ2 (7)

Total aerodynamic forces and moments are separated into rigid (i.e., body-fin 

surfaces in this report) and MAS (i.e., canard surfaces in this report). Rigid 

aerodynamic surface (RAS) forces include static (linear and nonlinear) and 

dynamic terms. Symbols in parenthesis indicate functional form of aerodynamic 

coefficients. The dynamic pressure is QQ = 1

2 ρρVV2 and aerodynamic reference area 

is SS = ππ

4

DD2, where DD is the projectile diameter and VV is the total velocity.

XXRR = −QQ �CCXX0(MM) + CCXXαα� (MM) sin αα� + CCXXαα�2 (MM) sin2 αα�

+ CCXXαα�3 (MM) sin3 αα� + CCXXαα�4 (MM) sin4 αα�� (8)

YYRR = −QQ �CCYY0(MM) + CCYYββ(MM) sin ββ + CCYYββ3 (MM) sin3 ββ

+ CCYYββ5 (MM) sin5 ββ� (9)

ZZRR = −QQ �CCZZ0(MM) + CCZZαα(MM) sin αα + CCZZαα3 (MM) sin3 αα

+ CCZZαα5 (MM) sin5 αα� (10)

RAS moments also include static (linear and nonlinear) and dynamic terms. The 

pitching moment accounts for a center-of-gravity (CCCCNN), which has been shifted 

from the center of gravity (CCCCNN,AA) used to obtain the aerodynamic data. The center 

of gravity is measured from the nose and is given in units of calibers.

LLRR = QQ �CCll0(MM) + CCllαα� (MM) sin αα� + CCll

αα�2 (MM) sin2 αα�

+ CCll

αα�3 (MM) sin3 αα� + CCll

αα�4 (MM) sin4 αα�

+ CCllpp(MM)

pp

2VV�

(11)
7

MMRR = QQ �CCmm0(MM) + CCmmαα(MM) sin αα + CCmmαα3 (MM) sin3 αα

+ CCmmαα5 (MM) sin5 αα + CCmmqq(MM)

qq

2VV�

− ZZRR�CCCCNN − CCCCNN,AA�DD

(12)

NNRR = QQ �−CCnn0(MM) − CCnnββ(MM) sin ββ − CCnnββ3 (MM) sin3 ββ

− CCnnββ5 (MM) sin5 ββ + CCmmrr(MM)

rr

2VV

�

+ YYRR�CCCCNN − CCCCNN,AA�DD

(13)

The following approach may be used to calculate MAS forces and moments for the 

th blade. First, compute local velocity at each blade from center-of-pressure data 

(CP, measured in calibers forward of CG), blade geometry (φφMMii

), and rigid-body 

states using the equation relating the velocity of 2 fixed points on a rigid body.

VV�⃑

MMii⁄II = VV�⃑CCCC⁄II + ωω��⃑BB⁄II × rr⃑CCCC→CCCCii (14)

where VV�⃑

CCCC⁄II = [uu vv ww]TT, ωω��⃑BB⁄II = [pp qq rr]TT, and rr⃑CCCC→CCCCii =

DD��CCCCXX�MM, δδCCii

� + CCCCNN − CCCCNN,AA� CCCCRRcccc �φφMMii� CCCCRRss �φφMMii��

TT

. The axial and 

radial center of pressure of the MAS is a function of Mach number and lifting 

surface deflection angle δδii.

Obtain local velocity at each blade in the blade coordinate system using the 

transformation matrix:

TT�⃑

BBMMii = �

1 0 0

0 cccc �φφMMii� ss �φφMMii�

0 −ss �φφMMii� cccc �φφMMii�

� (15)

⎣

⎢

⎢

⎡uuMMii

MMii

vvMMii

MMii

wwMMii

MMii

⎦

⎥

⎥

⎤

= TT�⃑

BBMMii �

uuMMii

BB

vvMMii

BB

wwMMii

BB

� (16)

Calculate local blade angle of attack from the local velocity in each blade 

coordinate system:

ααMMii = aa aaaa

⎣

⎢

⎢

⎡ wwMMii

MMii

�uuMMii

MMii

2

+ wwMMii

MMii

2

⎦

⎥

⎥

⎤

(17)
8

Determine lifting surface aerodynamic coefficients:

CCXX

MMii = CCXX0

MM (MM) + CCXXαα�MMiiMM (MM) sin αα�MMii + CCXXαα�MMii

2

MM (MM) sin2 αα�MMii

+ CCXXαα�MMii

3

MM (MM) sin3 αα�MMii + CCXXαα�MMii

4

MM (MM) sin4 αα�MMii

(18)

CCll

MMii = CCll0

MM(MM) + CCllαα�MMiiMM (MM) sin αα�MMii + CCll

αα�MMii

3

MM (MM) sin3 αα�MMii (19)

CCNN

MMii = CCNN0

MM (MM) + CCNNααMMiiMM (MM) sin ααMMii + CCNNααMMii

3

MM (MM) sin3 ααMMii

+ CCNNααMMii

5

MM (MM) sin5 ααMMii

(20)

CCmm

MMii = CCmm0

MM (MM) + CCmmααMMiiMM (MM) sin ααMMii + CCmmααMMii

5

MM (MM) sin5 ααMMii (21)

The nonlinearity with angle of attack of the canard aerodynamics is captured by 

using appropriate data based on the magnitude of angle of attack. For example, the 

coupled solution can provide the canard aerodynamic data that may require one fit 

for low angles of attack and another for higher angles of attack. 

Compute blade axial and normal forces and roll and pitching moments:

XXMMii = −QQMMiiSS CCXX

MMii (22)

LLMMii = QQMMiiSS ll

MMii (23)

NNMMii = −QQMMiiSS NN

MMii (24)

MMMMii = QQMMiiSS mm

MMii − NNMMii

�CCCCNN − CCCCNN,AA�DD (25)

Transform these blade forces and moments in the blade coordinate system to the

body-fixed coordinate system:

�

XXMMii

BB

YYMMii

BB

ZZMMii

BB

� = TT�⃑

BBMMii

TT �

XXMMii

0

NNMMii

� (26)

�

LLMMii

BB

MMMMii

BB

NNMMii

BB

� = TT�⃑

BBMMii

TT �

LLMMii

MMMMii

0

� (27)
9

This aerodynamic model only serves as a framework since the amount, source, and 

type of aerodynamic data (experimental or computational, coefficients or 

forces/moments), as well as flight phenomena such as interactions, dependence on 

aerodynamic roll angle, etc., are specific to a particular airframe at a given stage in 

development. New aerodynamic models may be required every time the 

configuration is changed.

5. Model Geometry and Grid

The projectile modeled in this study has a length of 427 mm and a main diameter 

of 83 mm. The computational model, as shown in Fig. 2, has a hemispherical nose, 

4 canards in front, and 10 fins at the back of the projectile. The main portion of the 

body consists of an 83-mm-diameter cylinder, which is followed by a conical 

boattail section. Two configurations, one with fins with a tab and the other with 

straight fins with no tabs, were modeled; however, the primary focus in this study 

has been the straight fins case. 

Fig. 2 Computational model of the projectile geometry

The computational grid for the main projectile body without the canards was 

created using MIME,

27 an unstructured mesh generator developed by Metacomp 

Technologies, Inc. GRIDGEN,

28 a commercially available software package, was 

used to generate structured meshes around the canards. The body alone 

computational mesh (Fig. 3) includes the body and the rear fins and consists of 

approximately 41 million cells. In general, most of the grid points were clustered 

in the boundary layer and the regions containing the nose canards and the afterbody 

fins. The boundary layer spacing normal to the wall was selected so that a wall 

function boundary condition could be used for turbulent flow. The first spacing near 

the wall was set at 0.25 mm on the body and 0.2 mm on the nose. A finer spacing 

of 3.5 × 10-3 mm was used on the fins and canards. These spacings resulted in y+ 

values of 20 or less on most of the body and a little higher of the order of 30 on the 

nose. The y+ values in the region containing the canards and the fins were 1.0 or 

less. The 4 nose canards were meshed separately and consisted of approximately 

1.3 million cells each. These 4 canard meshes were then overset with the 
10

background projectile body mesh using the Chimera procedure.

29 The final 

Chimera-overlapped mesh for each of the 2 (fin cant) models consists of 

approximately 43 million cells. A circumferential cross section (Fig. 4) shows full 

canard meshes. The outer boundaries of the canard meshes were chosen in a manner 

that required no cutting of the canard meshes by the main projectile body. However, 

the canards themselves were selected as cutters to cut holes in the background body 

mesh. A circumferential cut in the vicinity of the canards (Fig. 5) shows a crosssectional view of the merged background and canard meshes after the cutting 

process was finished. The Chimera procedure requires proper transfer of 

information between the background mesh and the canard meshes at every time 

step. However, the advantage is that the individual grids are generated only once 

and the Chimera procedure can then be applied repeatedly as required during the 

canard motion. There is no need to generate new meshes at each time step during 

the canard control maneuvers for roll, pitch, and roll-pitch-yaw. 

Fig. 3 Expanded view of the mesh in the XZ plane near the body

Fig. 4 View of the mesh in the circumferential plane, near the canard before overset
11

Fig. 5 View of the mesh in the circumferential plane, near the canards after overset cut

6. Results

The constraints-based coupled CFD/RBD approach was used in time-accurate, 

coupled computations using Navier-Stokes techniques for a canard-controlled, 

finned projectile at M = 0.6. Both unsteady aerodynamics and flight dynamics were 

computed simultaneously and the flight response of the projectile was determined. 

In all cases, full 3-D solutions were obtained and no symmetry was used. The idea 

of the coupled approach was to capture static and dynamic aerodynamic behavior 

over short time durations (of the order of 0.5 s) with different motions. Three 

different motions, pure pitch, pure roll, and a roll-pitch-yaw motion, were 

considered in this study, and coupled calculations were performed for aerodynamic 

characterization that included complexities due to high angle of attack such as flow 

interactions and unsteadiness. 

The analysis initially considers pure pitching and pure rolling motion before 

moving onto the coupled roll-pitch-yaw motion. This approach permits isolation of 

phenomena due to single DOF motion from full 3-D motion. Different DOF were 

enabled or disabled by setting the appropriate forces and/or moments in the 6DOF 

model to zero. In all cases, the Mach number is constant (M = 0.6). A time step of 

0.0002 s was used in the time-accurate coupled calculations. Smaller time steps 

were also tested and computed results were found to change insignificantly. The 

results in this report focus on airframe motion from nonequilibrium points without 

control inputs (i.e., canard deflections). In some instances the virtual model is 

released from appreciable angle of attack (15°) to investigate phenomena such as 

flow separation. For the pure roll, the roll rate of the virtual model starts at 5 Hz 

and decreases with time. Table 1 shows the initial conditions for the different 

coupled simulation cases. Initial conditions include all 12 state variables. Cases 1, 

2, and 3 represent the pure pitch, pure roll, and roll-pitch-yaw motion, respectively. 

Also, the canards are in the x orientation at the initial time (t = 0).
12

Table 1 Initial conditions for coupled simulations

x

(m)

y

(m)

Z

(m)

Φ 

(rad)

θ

(rad)

Ψ

(rad)

u

(m/s)

v

(m/s)

w

(m/s)

P

(rad/s)

q

(rad/s)

r

(rad/s)

Case1 

(pitch) 0.0 0.0 0.0 0.0 0.2618 0.0 198.0 0.0 53.1 0.0 0.0 0.0

Case 2 

(roll) 0.0 0.0 0.0 0.0 0.0 0.0 205.0 0.0 0.0 31.41593 0.0 0.0

Case 3

(rollpitchyaw)

0.0 0.0 0.0 0.0 0.2618 0.0 198.0 0.0 53.1 0.0 0.0 0.0

Some qualitative results are presented first for the roll-pitch-yaw motion (case 3). 

Figure 6 shows the particle traces emanating from the canards and moving 

downstream on the body and the fin region. The particle traces are shown at 3 

different instants in time, t = 0.03, 0.15, and 0.26 s. As the projectile goes through 

the roll-pitch-yaw motion, the aerodynamic angles change with time and so does 

the interaction of the particle traces on the afterbody fins. The particle traces in 

Figs. 6a, 6b, and 6c are representative of a typical high positive angle of attack, 

high negative angle of attack, and almost zero degree angle of attack, respectively. 

Figure 7 shows vorticity contours at 6 different instants in time during the very 

initial phase of this roll-pitch-yaw motion, 0.02 ≤ t ≤ 0.07 s. In this case, the 

projectile was released initially from an angle of attack of 15°. At time t = 0.02 s, 

the angle of attack is still close to 15°. As time increases, the angle of attack

decreases and is close to almost zero degree angle of attack at t = 0.07 s. Near zero 

or small angles of attack (t = 0.07 s), the outboard or tip canard vortices can be 

clearly seen to propagate straight down, whereas the inboard or root canard vortices 

are weak and can barely be seen at t = 0.07s. Specially, see the bottom canard facing 

the reader, the tip vortices move downstream and seem to hit one of the rear fins. 

As angle of attack is increased (time is decreased), the trajectory of the tip canard 

vortices changes, (i.e., it curves upwards over the length of the body, goes in 

between 2 rear fins at t = 0.05 s, and at high angles of attack of 15°, it hits the next 

rear fin [t = 0.02 s]). Also, it should be noted that root canard vortices, which are 

barely seen near zero degree angle of attack, can be seen at t = 0.05 s and very 

clearly at t = 0.04 s. At time, t ≤ 0.04 s, root canard body vortices grows in strength 

and size and also seem to interact with the crossflow vortices at higher angles of 

attack. The resultant flow field is quite complex with flow interactions that affect 

the forces and moments and are automatically included in the coupled 

computations. 
13

(a)

(b)

(c)

Fig. 6 Particle traces at 3 different instants in time during roll-pitch-yaw motion,

M = 0.6, (a) t = 0.03, (b) t = 0.15, and (c) t = 0.26 s
14

Fig. 7 Vorticity contours at different instants in time during initial phase of roll-pitch-yaw 

motion, M = 0.6, 0.02 ≤ t ≤ 0.07 s
15

Figures 8 and 9 show the corresponding computed surface pressure contours at the 

same instants in time during the initial stage of the roll-pitch-yaw motion, 0.02 ≤ t 

≤ 0.07 s. Figure 8 shows the surface pressures on the full body; high pressures are 

shown in red and low pressures in blue. Near almost zero or small angles of attack

(t = 0.07 s), the surface pressures are almost symmetric on the body other than the 

canards and fins, which can be clearly seen especially in the nose region. The 

surface pressure on each fin or canard is similar. Also, as expected, higher pressures 

in front of the nose, canards, and fins are evident. At earlier times, the angle of 

attack is as high as 15° (t = 0.02 s). With increasing angles of attack, the surface 

pressures become more and more asymmetric, with higher pressure on the windside and lower on the lee-side. Surface pressures on the body between the canards 

and the fins also get affected by canard root vortices. One can see this effect 

beginning at t = 0.06 s, which seems to get stronger at earlier times (t = 0.05 and 

0.04 s) and more pronounced at higher angles of attack (t = 0.03 and 0.02 s). Similar 

effects can also be observed near the afterbody fin region (Fig. 9). This figure shows 

an expanded view of the surface pressures on the afterbody. It shows large 

asymmetric pressure distribution on the fins at early times largely due to higher 

angles of attack and canard-fin flow interactions. As angle of attack is decreased to 

almost zero (t = 0.07 s), these effects are no longer present and the pressures on 

each fin looks the same.
16

Fig. 8 Surface pressure contours at different instants in time during initial phase of rollpitch-yaw motion, M = 0.6, 0.02 ≤ t ≤ 0.07 s
17

Fig. 9 Expanded view of surface pressure contours near the fins at different instants in time 

during initial phase of roll-pitch-yaw motion, M = 0.6, 0.02 ≤ t ≤ 0.07 s

6.1 Pure Pitching Case

Figure 10 shows the pitching motion history for the pure pitching simulation. The 

body is released from 15° and damps to less than 2° within about 3.5 cycles of 

motion. For these constrained motion simulations, the Euler pitch angle and angle 

of attack are equivalent. The pitch rate starts at zero, is phase shifted by 90° from 

the pitch angle, and features some sharper peaks than the pitch angle.

Fig. 10 Euler pitch angle (left) and pitch rate (right) for pure pitching case
18

The simulations compute the aerodynamic forces and moments acting on the body 

in addition to the angles and angular rates shown in Fig. 10. These data were 

analyzed to determine the aerodynamic coefficients at Mach 0.6. This technique, 

referred to as the direct method in this report, fit polynomials to the aerodynamic 

data as a function of angle of attack. Effects such as flow separation and vortex 

interactions were evident in most situations; therefore, the data were separated into 

low and high angle-of-attack regimes for analysis. Aerodynamic damping was 

appreciable in some instances (e.g., pitching moment). Damping contributions were 

removed to separate static and dynamic aerodynamics. The time history of the 

angular rates was used and the damping coefficient, based on reference diameter 

and velocity, was varied until hysteresis was removed in the curves of force or 

moment as a function of angle of attack. Semiempirical aeroprediction and virtual 

wind tunnel CFD techniques were used for the initial value of the damping 

coefficient. The final damping coefficient was then iteratively obtained when the 

total pitching moment (both static and dynamic components) matched computed 

total pitching moment.

Static aerodynamic forces and moments obtained from the direct method for the 

pure pitching simulation are provided in Figs. 11 and 12. In these figures, the data 

are presented with filled circles and the fits with lines. The low angle-of-attack data 

are given with blue circles and the high angle-of-attack data with green circles 

where applicable. The low angle-of-attack fits are a solid line and the high angleof-attack fits are a dashed line.
19

Fig. 11 Axial force coefficient of body-fin (top left), normal force coefficient of body-fin (top 

right), roll moment coefficient of body-fin (middle left), pitching moment coefficient of bodyfin (middle right), and side moment of body-fin (bottom) from pure pitching motion
20

Fig. 12 Axial force coefficient of canard (top left), normal force coefficient of canard (top 

right), roll moment coefficient of canard (bottom left), and pitching moment coefficient of 

canard (bottom right) from pure pitching

The computations integrated pressure distributions over specific surfaces to obtain 

the aerodynamic forces and moments. The static aerodynamics for the body and fin 

surfaces (i.e., no canards) are shown in Fig. 11. The axial force coefficient is 

relatively flat for angles of attack below 33°. The high angle-of-attack data possess 

some fluctuation above 8°, potentially due to transient effects, and were fit with an 

odd-order polynomial. All normal force data were captured with a single odd-order 

polynomial. The normal force is nearly linear with the angle of attack until about 

5° where the curve begins to fall off. The roll moment is small. In theory, the model 

does not possess roll inducing features (i.e., canted fins) so a nonzero roll moment 

indicates a truly 3-D flow or numerical asymmetries. At high angles of attack, there 

is a lot of scatter in the roll moment data suggesting unsteady phenomena. The 

pitching moment is strongly nonlinear with the angle of attack. The airframe is 

relatively less stable at higher angle of attack. Different odd-order polynomials 

were used for angles of attack below and above 8.7°. The fluctuation in the data for 

high angle of attack may again be attributed to transient effects. A small static side 

moment was evident in the data. A single curve was fit over all 
21

angles of attack since this modeling approach cannot reproduce the large scatter for 

angles of attack higher than about 9°. It is difficult to determine exactly what 

produces the scatter in the data at high angle of attack for static side moment.

The aerodynamic forces and moments on an individual canard are presented in 

Fig. 12. The data collected in the coupled simulations were transformed from 

airframe body coordinates to a local canard body coordinate system for analysis 

(i.e., the angle-of-attack ranges are not the same between Figs. 12 and 11). The 

canard-alone axial force coefficient is much smaller than the contribution to total 

axial force from the body and fins. The canard axial force decreases with angle of 

attack until about 6.5°. At higher angle of attack, a different flow state is present 

that increases the axial force of the canard with angle of attack. The normal force, 

roll moment, and pitching moment of the canard increases almost linearly up to 

6.5° angle of attack. Above 6.5°, stall reduces the effectiveness of the canards. A 

comparison of the fluctuating data with fits at high angle of attack highlights the 

difficulty in precisely modeling stall effects.

Tables 2 and 3 provide coefficients of the polynomial fits and relevant angle-ofattack ranges for the static aerodynamics of the body-fin and individual canard,

which were discussed and shown in Figs. 11 and 12 for the pure pitching 

simulation.

Table 2 Aerodynamic coefficients for body-fin at Mach 0.6 from direct analysis of pure 

pitching

Order Cx

(α < 3.3°)

Cx

(α ≥ 3.3°) CN

Ci

(α < 8.5°)

Cl

(α ≥

8.5°)

Cm

(α < 8.7°)

Cm

(α ≥ 8.7°) Cn

0 0.347577 0.334976 0 –0.001956 –0.01235 0 –0.05491 0

1 0 0.228077 11.23991 0 0.040174 –8.3319 –5.83862 –0.0399

2 0.067352 0 0 –0.098564 0 0 0 0

3 0 –0.7185 –119.323 0 0 235.9846 79.66208 –2.529

5 0 0 879.0171 0 0 –2985.24 –442.231 15

Table 3 Aerodynamic coefficients for canard at Mach 0.6 from direct analysis of pure 

pitching

Order Cx

(α < 6.5°)

Cx

(α ≥ 6.5°)

CN

(α < 6.5°)

CN

(α ≥ 6.5°)

Cl

(α < 6.5°)

Cl

(α ≥ 6.5°)

Cm

(α < 6.5°)

Cm

(α ≥ 6.5°)

0 0.003332 –0.06138 0 0.257626 0 –0.21496 0 0.38223

1 0 0.535632 1.048363 –1.45885 –0.8453 1.275679 1.529445 –2.18866

2 –0.90013 0 0 0 0 0 0 0

3 0 –5.08535 20.03176 15.87437 –13.7174 –14.8193 28.57757 23.76268

5 0 0 –1720.46 0 1200.296 0 –2455.96 0
22

In addition to the direct method, parameter estimation using a maximum likelihood 

method11,30–34 was conducted on the data for the pure pitching to obtain 

aerodynamic coefficients. This analysis assesses the aerodynamic model utility for 

use over a wider range of conditions and also permits more insight into the dynamic 

aerodynamic parameters. The aerodynamic coefficients from the direct method 

were used as the initial solution. The states measured from the CFD-RBD (black 

curves) and those calculated using the aerodynamic model (AM-RBD) in the 

parameter estimation (blue curves) are given in Fig. 13. The calculations faithfully 

follow the data; the root-sum-square fit error was 0.15°.

Fig. 13 States from parameter estimation analysis of pure pitching (Euler pitch angle rootsum-square fit error 0.15°)

Aerodynamic parameter estimation is often used in free-flight experiments, such as 

the spark range or onboard sensor technique, where flight motion (e.g., Euler 

angles, center-of-gravity position) is directly measured and the aerodynamic forces 

and moments required to produce that motion are inferred through analysis. In the 

CFD-RBD technique, the aerodynamic forces and moments are obtained in 

addition to the flight motions. This allows comparison of the measured and 

calculated aerodynamic forces and moments for different surfaces.

Moments that influence the pitching motion (e.g., pitching moment of body-fin, 

pitching moment of canards, and pitch damping moment of body-fin) are the only 

aerodynamic parameters that can be assessed using parameter estimation for the 

pure pitching. Figure 14 provides the time history of the pitching moment of the 

RAS (i.e., body-fin), MAS (i.e., canards), and total (i.e., body-fin and canards) for 

pure pitching. For about the first 0.05 s and a short timeframe near 0.15 s (as angle 

of attack remains above 8.7°), the pitching moment of the RAS is in the high angleof-attack regime. Different parameters were adjusted in the estimation algorithm 

for the pitching moment of the RAS in the low and high angle-of-attack regimes. 

This process yielded the discontinuity in the calculated pitching moment for the 
23

RAS near 0.05 s and underscores the challenge in modeling high angle-of-attack 

aerodynamics. A similar situation is encountered for the canard. The fluctuating 

data before 0.05 s and around 0.15 s indicates stall. The current analysis approach 

essentially fits through the fluctuation in the data, which changes the value of the 

higher-order terms in the aerodynamic model at or near the transition between low 

and high angle-of-attack coefficients. The correlation between unsteady 

fluctuations in the pitching moment of the body-fin and canards (e.g., before 0.05 s 

and around 0.15 s) and the fact that the canards are upstream of the body and fins 

suggest that fluctuations in the body-fin pitching moment are due to the interactions 

of stall-induced vortices shed off canards onto the body-fin pressure distribution. 

The body and finned surfaces could also be independently undergoing transient 

phenomena at these exact times/angles of attack but the visualizations do not 

support this claim. Further simulations could be conducted without canards to 

verify. The total pitching moment in Fig. 14 illustrates that this analysis captures 

appreciable nonlinearities, but the lack of agreement between measurements and 

calculations around 0.15 s highlights that more work remains in modeling high 

angle-of-attack aerodynamics (e.g., canard stall, vortex interactions).

Fig. 14 Pitching moments from parameter estimation analysis for pure pitching (top left = 

body-fin, top right = individual canard, bottom = total)
24

Table 4 summarizes the aerodynamic coefficients obtained from parameter 

estimation for the pure pitching. This technique yielded a pitch damping moment 

of –80.7 for the body-fins. The pitch damping of the canards is implicitly contained 

in the flight model for the canards.

Table 4 Aerodynamic coefficients at Mach 0.6 from parameter estimation of pure pitching, 

m = –80.7 C q

Order

Body-Fin Canard

Cm

(α < 10°)

Cm

(α ≥ 10°)

Cm

(α < 7.05°)

Cm

(α ≥ 7.05°)

0 0.0000 –0.3587 0.0000 0.1991

1 –8.033 –3.479 1.365 –0.269

3 240.28 49.22 14.96 0.00

5 –3505.5 –242.0 –990.7 0.0

6.2 Pure Rolling Case

The pure roll started with a 5-Hz roll rate. This allowed better characterization of 

the roll damping moment coefficient by increasing the relative contribution of roll 

damping to the dynamics. The measured and calculated roll angle and roll rate 

given in Fig. 15 agree well (root-sum-square fit error 0.1°). The roll rate decreases 

to near zero by 0.35 s and rotates about 155° in this time.

Fig. 15 States from parameter estimation analysis of pure rolling (Euler roll angle rootsum-square fit error 0.1°)

The roll moments are provided in Fig. 16 for the pure rolling motion. The largest 

roll moment is due to the roll damping of the RAS at higher roll rates. At later times 

when the roll rate is nearly zero, the small static roll moment is the main contributor 

to the RAS roll moment. The roll moment of the MAS is an order of magnitude 

smaller than the roll moment of the RAS over a majority of the time. There is some 

oscillation in the CFD-RBD data for times greater than about 0.4 s, but these values 

are very small.

 
25

Fig. 16 Rolling moments from parameter estimation analysis for pure rolling (top left = 

body-fin, top right = individual canard, bottom = total)

The aerodynamic coefficients from parameter estimation for pure rolling are 

provided in Table 5. The roll damping moment coefficient for the body-fin was 

calculated as –12.1.

Table 5 Aerodynamic coefficients at Mach 0.6 from parameter estimation of pure rolling,

l = –12.1 C p

Order Body-Fin Canard

Cl Cl

0 –0.00078 0

1 0 –0.695

3 0 –15.68

5 0 1251.2
26

6.3 Roll-Pitch-Yaw Case

Analysis of pure pitch and pure roll isolated the flight behaviors in those DOF. The 

roll-pitch-yaw case seeks to identify how coupling these DOF in a more realistic 

sense changes the flight response. The direct method was applied to the roll-pitchyaw flight data to determine static aerodynamic coefficients. These results are 

shown in Figs. 17 (body-fin) and 18 (canard). Direct method results for the pure 

pitching and the coupled roll-pitch-yaw are virtually identical. The only significant 

difference is for the aerodynamics of the canard at high angle of attack. The pure 

pitching predicts a slightly decreasing response in the coefficients poststall and the 

roll-pitch-yaw increases slightly with angle of attack.

Fig. 17 Axial force coefficient of body-fin (top left), normal force coefficient of body-fin (top 

right), roll moment coefficient of body-fin (bottom left), and pitching moment coefficient of 

body-fin (bottom right) from roll-pitch-yaw motion
27

Fig. 18 Axial force coefficient of canard (top left), normal force coefficient of canard (top 

right), roll moment coefficient of canard (bottom left), and pitching moment coefficient of 

canard (bottom right) from roll-pitch-yaw motion

The parameter estimation results for the roll-pitch-yaw are discussed next. 

Figure 19 shows favorable agreement between the measured and calculated roll, 

pitch, and yaw Euler angles. The root-sum-square fit errors were 0.32° for roll, 

0.19° for pitch, and 0.11° for yaw. Similar to the pure pitch, the airframe was 

initially pitched up to 15° and damps to about 2° within 4 cycles of motion. The 

small static roll moment, however, causes this 3DOF case to roll over 30°

(counterclockwise when viewed from behind airframe) in this time. As the body 

rolls, the yaw Euler angle increases. Additional motion out of the pitch plane is 

produced by the small static side moment.
28

Fig. 19 Euler roll (top left), pitch (top right), and yaw (bottom) angles from parameter 

estimation analysis of roll-pitch-yaw (root-sum-square fit errors of 0.32°, 0.19°, and 0.19° in 

Euler roll, pitch, and yaw angles)

Measured and calculated roll, pitch, and yaw body rates are given in Fig. 20. Similar 

to the pure pitching case, there is excellent agreement between the measured 

(coupled solution) pitch rate and the computed (or modeled) pitch rate. Note that 

the pitch rate is an order of magnitude higher than the roll and yaw rate. In general, 

the roll rate and the yaw rate also match reasonably well between the measurements 

and calculations as expected because the angles agree as favorably as shown in 

Fig. 19. There are some finely detailed, complex behaviors (e.g., roll rate) that the 

parameter estimation appears to capture. There are some discrepancies that can be 

observed near t = 0.15 s and are attributable to unsteady flow interaction effects. 
29

Fig. 20 Roll (top left), pitch (top right), and yaw (bottom) rates from parameter estimation 

analysis of roll-pitch-yaw

The total (body-fin and canard) aerodynamic moments that yield the kinematics 

and dynamics shown in Figs. 19 and 20 are provided in Fig. 21. Again, roll and yaw 

moments are an order of magnitude lower than pitch moments. The pitch moment 

illustrates a high degree of nonlinearity as discussed for the pure pitching case. The 

fluctuation in the pitch moment prior to 0.05 s and around 0.15 s again suggests 

unsteadiness due to canard stall. This fluctuation is also apparent in the roll and 

yaw moments at these times. A canard-on-body-fin flow interaction is offered to 

explain fluctuation in body-fin aerodynamics (not shown for roll-pitch-yaw) during 

periods of canard stall and is also supported by the flow visualization, but this 

hypothesis needs to be tested with further analysis. Capturing these high angle-ofattack phenomena is critical to understanding the flight behaviors since these 

fluctuations in the aerodynamic moments propagate through the dynamics and 

emerge as the sharp nonlinearities, for example, as seen in the roll rate and yaw rate 

around 0.15 s in Fig. 20.
30

Fig. 21 Total moments from parameter estimation analysis for roll-pitch-yaw (top left =

roll, top right = pitch, bottom = yaw)

Figure 22 compares the aerodynamic moments obtained from the direct method and 

parameter estimation (numerical values provided in the Appendix). The 2 methods 

reasonably agree. Some discrepancies can be observed, for example, in canard roll 

moment and especially at higher angles of attack. Models underpinning both 

datasets clearly need to be refined to better account for unsteady high angle-ofattack aerodynamics. Overall, the data provided by the pure pitch and pure roll 

analysis yield a reasonable prediction of the coupled roll-pitch-yaw behavior. This 

is supported by the agreement between the results in Fig. 22 for the parameter 

estimation and direct method (i.e., input to parameter estimation routine). Capturing 

the rapidly fluctuating aerodynamic forces and moments on the afterbody-fins due 

to vortex interactions during time periods of canard stall appears to be the prime 

motivator for performing roll-pitch-yaw simulations.
31

Fig. 22 Rolling moment coefficient of body-fin (top left), pitching moment coefficient of 

body-fin (top right), side moment coefficient of body-fin (middle), rolling moment coefficient 

of canard (bottom left), and pitching moment coefficient of canard (bottom right) from 

direct method and parameter estimation of roll-pitch-yaw

7. Conclusions

New constraints-based computations were performed on a canard-controlled finned 

projectile using a coupled CFD/RBD procedure and a 3-D, unsteady, unstructured 

Navier-Stokes technique. In the coupled calculations, both unsteady aerodynamics 

and flight dynamics were computed simultaneously and the response of the 

projectile was determined in pure pitch, pure roll, and roll-pitch-yaw motions. A 

full aerodynamic model, including separate body-fin and canard contributions for 
32

this configuration, was generated from the coupled solutions using parameter 

estimation algorithms and a direct method. Comparison of the coupled solutions 

and the aerodynamic model shows generally good agreement indicating that the 

aerodynamic model sufficiently characterizes flight motions for this configuration 

with the exception of high angle of attack. Computed flow fields show unsteady 

vortices during canard stall that yield canard-on-afterbody-fin flow interactions at 

high angles of attack resulting in unsteady aerodynamic forces/moments. It is 

important to note that these unsteady effects are automatically taken into account 

in the coupled approach. Aerodynamic models for this situation, potentially 

including the stochastic nature of the aerodynamic forces and moments during high 

angle of attack, need to be improved. In addition, these results suggest that isolating 

the pitch and roll dynamics and using this analysis to predict the coupled roll-pitchyaw motion is appropriate for low angle of attack. At higher angle of attack, the 

unsteadiness and interactions may require more sophisticated analysis techniques.

This approach based on new constraints-based coupled fly-out simulations to 

generate aerodynamics for pitch, roll, and pitch-roll-yaw flights shows good 

promise. The same constrained-based coupled fly-out simulation approach can be 

easily extended for aerodynamic/flight dynamic characterization with canard 

deflections. For guided flights with deflecting canards, control algorithms need to 

be developed and implemented in coupled CFD/RBD simulations. Ongoing efforts 

are focused on coupled calculations for pitch, roll, and roll-pitch-yaw control 

maneuvers with canard deflections, both open-loop and closed-loop. These results 

will illustrate the utility of high fidelity modeling of the flight physics in the control 

formulation. 
33

8. References

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2. Moore FG, Moore LY. Approximate method to calculate nonlinear rolling 

moment due to differential fin deflection. Journal of Spacecraft and Rockets.

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3. Fresconi FE, DeSpirito JS, Celmins I. Flight performance of a small diameter 

munition with a rotating wing actuator.Journal of Spacecraft and Rockets. Vol. 

2015;42(2):305–319.

4. Fresconi FE, Celmins I, Ilg M, Maley J. Projectile roll dynamics and control 

with a low-cost skid-to-turn maneuver system. Journal of Spacecraft and 

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5. Dupuis A, Hathaway W. Aeroballistic range tests of the basic finner reference 

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Appendix. Aerodynamic Coefficients at Mach 0.6
38

Table A-1 Aerodynamic coefficients at Mach 0.6 from direct and parameter estimation 

analysis of roll-pitch-yaw

O

r

d

e

r

Body-Fin CCll Canard CCll (low αα) Canard CCll (high αα) Body-Fin CCnn

Direct Parameter 

Estimation

Direct 

(α < 6.3°)

Parameter 

Estimation 

(α < 7.05°)

Direct 

(α ≥ 6.3°)

Parameter 

Estimation

(α ≥ 7.05°)

Direct Parameter 

Estimation

0 –0.00126 –0.00078 0 0 –0.064654 –0.0867 0 0

1 0 0 –0.84469 –0.695 –0.179914 0.083591 –0.03991 –0.03991

3 0 0 –13.9344 –15.68 0 0 –2.52907 –2.52907

5 0 0 1229.269 1251.2 0 0 14.61258 14.61258

O

r

d

e

r

Body-Fin CCmm (low αα) Body-Fin CCmm (high αα) Canard CCmm (low αα) Canard CCmm (high αα)

Direct 

(α < 8.7°)

Parameter 

Estimation

(α < 10°)

Direct 

(αα ≥ 88. 77oo)

Parameter 

Estimation 

(αα ≥ 1111)

Direct 

(αα < 66. 33oo)

Parameter 

Estimation 

(αα <

77. 00oo)

Direct 

(αα ≥ 66. 33oo)

Parameter 

Estimation 

(αα ≥

77. 00oo)

0 0 0 –0.08071 –0.3587 0 0 0.134338 0.1991

1 –8.33163 –8.033 –5.64646 –3.479 1.52846 1.365 0.148636 –0.269

3 236.0836 240.28 77.98839 49.22 28.79212 14.96 14.41674 0

5 –2997.08 –3505.5 –438.128 –242 –2465.12 –990.7 0 0
39

List of Symbols, Abbreviations, Acronyms, and Nomenclature

3-D 3-dimensional

CFD computational fluid dynamics

DOF degrees of freedom

MAS moveable aerodynamic surfaces

RAS rigid aerodynamic surfaces

RBD rigid body dynamics

φφ, θθ, ψψ Euler roll, pitch, and yaw angles, rad

II moment-of-inertia tensor, kg m2

mm mass, kg

DD diameter, m

SS = ππ

4

DD2 reference area, m2

VV velocity, m/s

MM Mach number

αα, ββ, αα�,φφAA

pitch angle of attack, yaw angle of attack, total angle of 

attack, aerodynamic roll angle, rad

ρρ atmospheric density, kg/m3

qq� = 1

2 ρρVV2 dynamic pressure, Pa

xxBB, yyBB, zzBB body-fixed coordinate system 

XX, YY, ZZ force components, N

LL, MM, NN moment components, Nm

CCXX, CCYY, CCZZ axial, side, and normal force coefficients

CCll, CCmm, CCnn roll, pitch, and side moment coefficients

CCmmqq pitch damping moment coefficient

CCllpp roll damping moment coefficient
40

CCCCRR radial center-of-pressure, calibers

CCCCXX axial center-of-pressure, calibers

CCCCNN center-of-gravity, calibers

NN number

δδ deflection, rad

VV�⃑

CCCC⁄II = [uu vv ww]TT velocity of projectile center-of-gravity, m/s

ωω��⃑BB⁄II = [pp qq rr]TT angular velocity of projectile, rad/s

TT�⃑

BBMMii

transformation matrix from body frame to ith lifting surface 

frame

tt time, s

WW, FF, , HH conservative variables, inviscid flux vector, viscous flux 

vector, source term

VV, AA cell volume, cell area

subscripts i

th lifting surface

0, 1, 2, 3, 5 zeroth, first, second, third, fifth order terms

MM maneuver surface, measurement
41

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