Navier-Stokes Predictions of the Individual Components 

of the Pitch-Damping Coefficient Sum 

by Paul Weinacht 

ARL-TR-3169 April 2004 

Approved for public release; distribution is unlimited. 
NOTICES 

Disclaimers 

The findings in this report are not to be construed as an official Department of the Army position unless 

so designated by other authorized documents. 

Citation of manufacturer’s or trade names does not constitute an official endorsement or approval of the 

use thereof. 

Destroy this report when it is no longer needed. Do not return it to the originator. 
Army Research Laboratory 

Aberdeen Proving Ground, MD 21005-5066 

ARL-TR-3169 April 2004 

Navier-Stokes Predictions of the Individual Components 

of the Pitch-Damping Coefficient Sum 

Paul Weinacht 

Weapons and Materials Research Directorate, ARL 

Approved for public release; distribution is unlimited. 
 ii

Report Documentation Page Form Approved 

OMB No. 0704-0188

Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and 

maintaining the data needed, and completing and reviewing the collection information. Send comments regarding this burden estimate or any other aspect of this collection of information, 

including suggestions for reducing the burden, to Department of Defense, Washington Headquarters Services, Directorate for Information Operations and Reports (0704-0188), 1215 Jefferson 

Davis Highway, Suite 1204, Arlington, VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to any penalty for failing to 

comply with a collection of information if it does not display a currently valid OMB control number. 

PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS. 

1. REPORT DATE (DD-MM-YYYY)

April 2004 

2. REPORT TYPE 

Final 

3. DATES COVERED (From - To) 

1994–2003 

5a. CONTRACT NUMBER 

5b. GRANT NUMBER 

4. TITLE AND SUBTITLE 

Navier-Stokes Predictions of the Individual Components of the Pitch-Damping 

Coefficient Sum 

5c. PROGRAM ELEMENT NUMBER

5d. PROJECT NUMBER 

1L1612618AH80 

5e. TASK NUMBER 

6. AUTHOR(S) 

Paul Weinacht 

5f. WORK UNIT NUMBER 

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 

U.S. Army Research Laboratory 

ATTN: AMSRD-ARL-WM-BC 

Aberdeen Proving Ground, MD 21005-5066 

8. PERFORMING ORGANIZATION 

 REPORT NUMBER 

ARL-TR-3169 

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR'S ACRONYM(S) 

11. SPONSOR/MONITOR'S REPORT 

 NUMBER(S) 

12. DISTRIBUTION/AVAILABILITY STATEMENT 

Approved for public release; distribution is unlimited. 

13. SUPPLEMENTARY NOTES 

14. ABSTRACT 

An approach for predicting the two individual aerodynamic damping coefficients that form the pitch-damping coefficient sum is 

presented. The coefficients are obtained using prescribed or forced motions that independently excite the two different angular 

rates that are associated with the two damping coefficients. A key feature of the approach is that steady flow fields are produced 

by the selected motions. Steady flow computational fluid dynamics approaches can be applied, allowing results to be obtained 

in a computationally efficient manner. Application of the technique is made to an axisymmetric projectile configuration. The 

predicted pitch-damping coefficient sum obtained by adding the individually determined coefficients is in excellent agreement 

with previous predictions of the pitch-damping coefficient sum and with experimental data. Additional validation of the 

approach is obtained through comparisons with prior results from numerical solutions of the nonlinear unsteady potential 

equation. The individual coefficients are also compared with slender body theory, and the results show similar trends though the 

slender body theory appears to underpredict the various coefficients. 

15. SUBJECT TERMS

aerodynamics, projectile flight mechanics, pitch-damping 

16. SECURITY CLASSIFICATION OF: 

19a. NAME OF RESPONSIBLE PERSON

Paul Weinacht 

a. REPORT 

UNCLASSIFIED 

b. ABSTRACT 

UNCLASSIFIED

c. THIS PAGE 

UNCLASSIFIED

17. LIMITATION 

OF ABSTRACT 

UL 

18. NUMBER 

OF PAGES 

44 19b. TELEPHONE NUMBER (Include area code) 

410-278-4280 

Standard Form 298 (Rev. 8/98) 

Prescribed by ANSI Std. Z39.18
 iii

Contents 

List of Figures iv

List of Tables v

1. Introduction 1

2. Theoretical Background 2

2.1 The Force and Moment Expansion .................................................................................2 

2.2 Planar Motions ................................................................................................................4 

2.3 Helical Motions ...............................................................................................................7 

2.4 CG Translation Relations ..............................................................................................11 

3. Computational Approach 11

4. Results 15

5. Conclusion 27

6. References 28

List of Abbreviations and Symbols 31

Distribution List 35 
 iv

List of Figures 

Figure 1. Schematic of ANSR configuration..................................................................................2 

Figure 2. Angular orientation of projectile relative to velocity vector and earth-fixed 

coordinate frame. .......................................................................................................................5 

Figure 3. Schematic of planar looping motion................................................................................6 

Figure 4. Helical motion with nonzero α

~

& and zero q

~ ...................................................................8 

Figure 5. Vertical projection of helical motion with nonzero α

~

& and zero q

~ . ...............................8 

Figure 6. Helical motion with zero α

~

& and nonzero q

~ ...................................................................9 

Figure 7. Vertical projection of helical motion with zero α

~

& and nonzero q

~ . ...............................9 

Figure 8. Variation of aerodynamic moment Cm with angular velocity, q = 0 helical motion, 

M = 2.5, ANSR, sin–1 δ = 2°...................................................................................................16 

Figure 9. Variation of aerodynamic moment Cm with sine of the angle of attack, q = 0 helical 

motion, M = 2.5, ANSR, V

ΩD .................................................................................................17 

Figure 10. Variation of aerodynamic moment Cm with angular velocity, α& = 0 helical 

motion, M = 2.5, ANSR, β = 2°.............................................................................................18 

Figure 11. Variation of aerodynamic moment Cm with rotational velocity ratio, α& = 0 helical 

motion, M = 2.5, ANSR, 0.010

V

D = Ω . ...................................................................................18 

Figure 12. Variation of damping moment coefficients with CG location, M = 2.5, 

ANSR, L/D = 5. .......................................................................................................................20 

Figure 13. Variation of damping moment coefficients with CG location, M = 2.5, 

ANSR, L/D = 7. .......................................................................................................................20 

Figure 14. Variation of damping moment coefficients with CG location, M = 2.5, 

ANSR, L/D = 9. .......................................................................................................................21 

Figure 15. Variation of damping force coefficients with CG location, M = 2.5, ANSR, 

L/D = 5.....................................................................................................................................21 

Figure 16. Variation of damping force coefficients with CG location, M = 2.5, ANSR, 

L/D = 7.....................................................................................................................................22 

Figure 17. Variation of damping force coefficients with CG location, M = 2.5, ANSR, 

L/D = 9.....................................................................................................................................22 

Figure 18. Variation of damping moment coefficient sum with CG location, M = 2.5, 

ANSR, L/D = 5. .......................................................................................................................23 

Figure 19. Variation of damping moment coefficient sum with CG location, M = 2.5, 

U.S. ANSR, L/D = 7. ...............................................................................................................23 

Figure 20. Variation of damping moment coefficient sum with CG location, M = 2.5, 

ANSR, L/D = 9. .......................................................................................................................24 
 v

Figure 21. Variation of the pitch-damping moment coefficients with Mach number, 

ANSR, L/D = 5, middle CG location.......................................................................................25 

Figure 22. Variation of the pitch-damping moment coefficient sum with Mach number, 

ANSR, L/D = 5, middle CG location.......................................................................................26 

Figure 23. Development of pitch-damping force coefficients over ANSR body, M = 2.5, 

L/D = 5, middle CG position. ..................................................................................................26 

Figure 24. Development of pitch-damping moment coefficients over ANSR body, M = 2.5, 

L/D = 5, middle CG position. ..................................................................................................27 

List of Tables 

Table 1. CG location for ANSR flight bodies...............................................................................15 
 vi

INTENTIONALLY LEFT BLANK. 
 1

1. Introduction 

Traditionally in aeroballistic applications, the pitch-damping coefficient has been treated as the 

sum of two individual coefficients that produce an aerodynamic moment proportional to the 

angular rate associated with the angle of attack. In fact, these two individual coefficients 

represent moments proportional to two different angular rates, although for many 

nonmaneuvering flight trajectories, including those flown in ballistic aerodynamic ranges, these 

two angular rates are essentially equivalent. For this reason, the pitch-damping coefficient sum 

is often treated as a single parameter. In some cases, such as for maneuvering flight vehicles, the 

simplification that the two angular rates are equivalent is no longer valid, and the two individual 

components of the pitch-damping coefficient sum must be determined independently. 

Experimental determination of these coefficients is difficult at best and is probably not possible 

using traditional aerodynamic ranges. 

Prior research has focused on the development and application of computational methods for 

predicting the pitch-damping coefficient sum. The basis of the technique is to impose a 

particular motion (coning motion) on the flight vehicle, which produces moments that are 

proportional to the pitch-damping coefficient sum. Because coning motion produces a steady 

flow field in the cases of interest, efficient numerical techniques can be readily applied and the 

pitch-damping coefficient sum determined in a cost-efficient manner. These techniques have 

been benchmarked with experimental data, and good agreement between computation and 

experiment has been found for a number of flight vehicle geometries (1–10). 

In the current effort, the method is extended so that the individual components of the pitchdamping coefficient can be determined independently. The method uses imposed motions that 

excite the two angular rates independently so that the forces and moments attributable to these 

rates can be assessed separately. A key feature of the approach is that the motions produce 

steady flow fields that can be computed in a cost-efficient manner. The techniques described 

here represent a unique aerodynamic capability for a problem that is difficult to address 

experimentally. Other than approximate methods, these results most likely represent the first 

numerical predictions of the individual components of the pitch-damping coefficient sum for 

flight vehicles. Subsequent to its original publication (8–10), the technique has also been 

adopted by other researchers (11). 

This report presents a derivation of the transverse aerodynamic force and moment equations 

associated with the motions of interest using a general force and moment expansion for 

symmetric flight bodies. The resulting equations demonstrate that the aerodynamic forces and 

moments can be excited independently using the appropriate motions. Results are presented for 

a family of axisymmetric flight bodies at supersonic flight velocities. A schematic of the ArmyNavy spinner rocket (ANSR) flight body geometries is shown in figure 1. Both of the 
 2

Figure 1. Schematic of ANSR configuration. 

components ( mq

C and mα& C ) of the pitch-damping coefficient sum, mq mα& C + C , are determined 

independently. Comparison of the predicted aerodynamic coefficients is made with slender body 

theory and numerical solutions of the nonlinear unsteady potential equation. The sum of the two 

individually determined coefficients is also compared with the pitch-damping coefficient sum, 

mq mα& C + C , determined from coning motion. Both methods of determining the pitch-damping 

coefficient sum are in good agreement with experimental data. 

2. Theoretical Background 

2.1 The Force and Moment Expansion 

It is common in many aerodynamics applications to use a body-fixed, nonrolling coordinate 

system to describe both the dynamics and the system of forces and moments that act on the flight 

vehicle (12). The nonrolling coordinate system allows the description of the vehicle dynamics to 

be simplified for certain classes of flight vehicles that possess particular types of geometric 

symmetry. Rotationally symmetric flight vehicles, which are the focus of the current research, 

represent one class of vehicles where the nonrolling frame has been effectively (and 

traditionally) used. For more complicated geometries, such as aircraft, the advantages of the 

nonrolling frame are reduced and other coordinate frames such as a completely body-fixed 

coordinate system are typically used. 
 3

In the current effort, the primary reason for initially describing the aerodynamic forces and 

moments using the nonrolling coordinate system is the fact that the description is well 

established for symmetric flight vehicles. The nonrolling coordinate frame is an orthogonal 

right-handed system ) z

~ y, ~ x, ~( centered at the body’s center of gravity (CG). The x~ -axis is 

aligned along the projectile longitudinal axis with the positive direction oriented toward the 

projectile nose. The z

~ -axis is “initially” oriented downward with the x~ - z~ plane perpendicular 

to the ground. The angular motion of the nonrolling coordinate frame is such that, with respect 

to an inertial frame, the x

~ -component of the coordinate frame’s angular velocity is zero. 

Although the time-dependent orientation of the nonrolling frame may be difficult to visualize, 

the nonrolling frame is essentially equivalent to the “fixed-plane” coordinate system for small 

amplitude motions. In the fixed-plane coordinate system, the x

~ - z~ plane remains perpendicular 

to the ground for all time. The total angular velocity of the flight vehicle can be described in 

terms of its angular velocity components r) ~ q, ~ ( along the x p, ~ , y~ , and z~ axes, respectively. 

The angular velocity of the nonrolling frame can be described in terms of the transverse angular 

velocities q

~ and r~ because the angular velocity of the nonrolling frame along the x

~ -axis is 

always zero. The flight body may, however, have a nonzero spin rate, p, about its longitudinal 

axis. Further details about these coordinate frames are discussed in reference (12). 

The moment expansion for a rotationally symmetric missile in the nonrolling coordinate frame is 

shown in equation 1. This moment expansion is similar to the moment proposed by Murphy 

(12). The moment formulation uses complex variables to separate the moment components, Cm

~

and Cn

~ , that are oriented along the y

~ and z~ axes, respectively. The third moment component, 

the roll moment, can be handled separately and is not of consequence in this study. 

ξ μ ξ α α α + − ′ 



 



  − 

 



 + = ~ iC ~ C ~ C iC

V

p C

~ C i ~

m n np m mq m &

l . (1) 

In the moment expansion, the pitching moment coefficient slope, mα C , and the coefficient 

mα& C represent moments that are proportional to the complex yaw, ξ

~ , and yawing rate, ξ ′ ~ , 

respectively. 

The complex yaw and yawing rate are defined as follows. (In the analysis presented here, there is 

no need to distinguish between pitch and yaw, and the terms may be interchanged.) The usage 

follows that of Murphy (12). 

V

w

~ v i ~ ~ + ξ = , (2) 

and 





 



 ′ =

l

s d

~ ~ dξ ξ . (3) 
 4

Here, v

~ and w~ are the y~ - and z~ -components of the velocity vector, V∞

r , that describes the 

velocity of the body CG relative to the inertial frame. The magnitude of this vector is denoted as 

V. The angular rate is obtained by taking the derivative of the complex yaw with respect to the 

flight path coordinate, s, which is nondimensionalized by some characteristic length, l , typically 

the body diameter. 

The coefficient, mq C , represents a moment that is proportional to the complex transverse angular 

velocity of the vehicle, μ

~ , as defined in the following: 

( )

V

r

~ q i ~ ~ + l

μ = . (4) 

Here, q

~ and r~ are the y

~ - and z~ -components of angular velocity of the vehicle in the 

nonrolling coordinate system. The remaining coefficient in the moment expansion, the Magnus 

moment coefficient, npα C , accounts for a side moment due to flow asymmetries produced by the 

combination of spin and yaw. 

It should be noted that the moment formulation neglects the variation of the moments with roll 

angle under the assumption that these variations are small. For axisymmetric vehicles, the 

variations with roll angle should not exist because the geometry will not change as the roll 

orientation changes. Roll variations in the aerodynamic coefficients for other types of 

rotationally symmetric vehicles are typically negligible for small amplitude motions. In general, 

roll variations may be difficult to detect in flight because the effect of roll orientation tends to be 

averaged out over the course of a yaw cycle if the body is spinning. 

2.2 Planar Motions 

For planar motions, the aerodynamic moments that act on an axisymmetric flight body can be 

written in terms of the following expansion. The force expansion has a similar form. 





 



  +



 



 + 



 



  + 

 



 + = V

C

V

q C C sin C

V

p C i ~ C i ~

m n np m mq m

l l &l

&

α

α γ α α α . (5) 

Two damping moments, mα& C and mq C , represent moments proportional to the angular rates α&

and q that are associated with the angle of attack, α, and the angular displacement of the 

longitudinal axis of the body with respect to the earth-fixed axis system denoted by the angle, θ , 

respectively. These angles are displayed schematically in figure 2 for the case of planar motion. 

For a typical ballistic trajectory, over the course of many yaw cycles, the flight body will travel a 

nearly rectilinear flight path. In this case, the free-stream velocity vector, V∞

r , has a fixed 

orientation with respect to the earth-fixed axes, xe and ze. For a rectilinear flight path, the 

angular rates α& and q will be equal. The moment expansion can be simplified by combining the 

two damping coefficients into a single coefficient sum, which is proportional to a single angular 
 5

Figure 2. Angular orientation of projectile relative to velocity vector and earth-fixed coordinate frame. 

rate (either α& or q). For maneuvering flight bodies, this simplification may not be valid, and the 

individual coefficients must be treated independently. 

To produce aerodynamic moments proportional to the damping terms, one or both of the angular 

velocities must be nonzero. One simple planar motion that produces a nonzero q and zero α&

angular velocity is a circular looping motion as shown in figure 3. If such a motion is performed 

at constant angular velocity, Ω , ) (q = θ

& = Ω , and constant angle of attack (which may or may 

not be zero), the following form of the force and moment expansions results: 





 



 Ω

+ 



 



  + 

 



 + = V

C C sin C

V

p C i ~ C i ~

m n np m mq

l l

α α α . (6) 

Here, the in-plane moment, Cm , contains contributions from both the pitching moment slope, 

mα C , and the pitch-damping coefficient, mq C . The side moment, Cn, is identical in form to the 

side moment due to constant α motion and is independent of the angular rate q. Because the 

damping terms are independent of the spin rate and the side moment is independent of the 

angular rate q, the spin rate is assumed to be zero to simplify the discussion of the looping 

motion that follows. 

With respect to the inertial frame of reference, the flow field is periodic and unsteady. However, 

in the nonrolling frame (shown in figure 3 as the x~ – z~ axes), the flow field is potentially steady. 

Indeed, the moment expansion displays no unsteadiness since all the terms on the right-hand side 

are constants. Because the flow field is potentially steady in the nonrolling frame for this type of 

motion, this frame is suitable for use with computational approaches that are based on steady 

flow techniques. 
 6

Figure 3. Schematic of planar looping motion. 

For looping motion, the moment expansion contains contributions from two of the aerodynamic 

coefficients. The contribution from the pitching moment coefficient slope, mα C , can be 

eliminated if the body longitudinal axis is tangent to the radius of curvature of the loop. In this 

case, the angle of attack is zero, and the damping moment, mq C , is directly proportional to the 

net aerodynamic moment. If the damping moment exhibits a dependence on angle of attack, 

several experiments or computations involving different looping rates must be performed for 

each angle of attack. (If the looping rate is varied, the loop radius must also be changed to 

maintain a constant vehicle velocity: V = ΩRo .) The damping moment can be obtained by 

computing the variation in the net aerodynamic moment with looping rate. If the net 

aerodynamic moment displays a nonlinear variation with both angle of attack and looping rate, it 

may be difficult to separate the contributions attributable to the mα C and mq C , without 

additional information or assumptions. In general, variations of the coefficients with angle of 

attack might be reasonably expected, while variations in the coefficients with angular rate are 

less common. In any event, the looping motion can be used to check for these types of nonlinear 

variations. 

One feature of this type of motion is that high angular velocities or large loop radii are required 

to generate high velocities, making this type of motion impractical for use in experimental 

testing. However, this is not a problem for computational approaches because large loop radii or 

high angular velocities can be easily accommodated. 
 7

2.3 Helical Motions 

Other more complicated motions can be proposed, which produce aerodynamic moments 

proportional to the angular rates, μ

~ and ξ ′ ~ , while still producing steady flow fields when 

viewed from the appropriate coordinate frame. Two such motions require the CG of the flight 

vehicle to traverse a helical flight path. The first motion requires the vehicle’s longitudinal axis 

to be oriented in the same direction as the center of rotation of the helix but displaced by a 

constant distance. Figure 4 shows a three-dimensional (3-D) view of the motion. A twodimensional (2-D) projection of this motion on the vertical plane is shown in figure 5. This 

particular motion produces no rotation of the nonrolling coordinate frame relative to an earthfixed coordinate frame, and hence, the angular velocity μ

~ is zero. The angle of attack and its 

angular rate vary continuously, producing moment components associated with the coefficients 

mα C and mα& C , respectively. This motion is referred to as “q = 0 helical motion” because the 

angular rates associated with the damping coefficient mq

C are zero. 

For the second motion, the longitudinal axis of the flight vehicle remains tangent to the helical 

flight path at each point along the trajectory. Figure 6 shows a 3-D view of this motion. A 2-D 

projection of this motion on the vertical plane is also shown in figure 7. The angle of attack of 

the incident airstream is zero because both the longitudinal axis of the body and the free-stream 

velocity vector are tangent to the flight path. The resulting yawing rate is also zero because the 

angle of attack is constant. The angular orientation of the flight body changes continuously with 

respect to an earth-fixed reference frame, producing a nonzero angular rate, μ

~ . As a result, 

moment components associated with the damping moment mq C are produced. This motion is 

referred to as “α& = 0 helical motion” because the angular rates associated with the damping 

coefficient mα& C are zero. The α& = 0 helical motion produces a steady flow field when viewed 

from the appropriate coordinate system, while its 2-D projection is clearly time dependent. 

The net transverse aerodynamic moment in the nonrolling frame can be determined for each of 

the motions by substituting the appropriately determined angle of attack, ξ

~ , and angular rates, 

ξ ′ ~ and μ

~ , into equation 1. The angle of attack and angular rates for both types of helical 

motion are shown in equations 7–14. 

q = 0 Helical Motion 

i t e

V

w

~ v i ~ ~ Ω = − + ξ ≡ δ , (7) 

V

R

U R

R o

2

o

2 2

o Ω =

+ Ω

Ω δ = , (8) 
 8

Figure 4. Helical motion with nonzero α

~

& and zero q

~ . 

Figure 5. Vertical projection of helical motion with nonzero α

~

& and zero q

~ . 

i t e

V

i ~ Ω 



 



 Ω ξ ′ = − δ l , (9) 

and 

 0 ~

μ = . (10) 

α& = 0 Helical Motion

( ) ( ) β μ β i t cos sin e

V V

r

~ q i ~ ~ Ω Ω = + ≡ l l , (11) 
 9

Figure 6. Helical motion with zero α

~

& and nonzero q

~ . 

Figure 7. Vertical projection of helical motion with zero α

~

& and nonzero q

~ . 

V

U

cos β = , (12) 

V

R sin oΩ β = , (13) 

and 
 10

 0 ~ ~ ξ = ξ ′ = . (14) 

Here, Ω is the angular velocity of the body about the helix axis, Ro is the perpendicular distance 

between the helix axis and the body CG, U is the component of velocity along the helix axis, and 

V is the total linear velocity of the CG. 

To completely define the motions, the spin rate of the body p needs to be specified. It is 

convenient to set the spin rate of the body p to zero because this will eliminate the contribution to 

the net aerodynamic moment from the Magnus moment, although this is not required for the 

α& = 0 helical motion because the complex angle of attack is also zero. Zero spin rate is obtained 

by the addition of a second angular velocity oriented along the flight body’s longitudinal axis 

that is equal and opposite of the longitudinal component of the angular velocity required to 

produce the rotation about the axis of the helix. This second angular velocity has no effect on 

the transverse angular velocities q

~ and r. ~

For each of the helical motions, the transverse aerodynamic moment in the nonrolling frame will 

be periodic in time, which also indicates that the flow field will be periodic in time when viewed 

from the nonrolling coordinate frame. However, for both types of helical motion, a steady flow 

field should be observed when examined from an orthogonal right-handed coordinate system that 

has its x-axis aligned with the longitudinal axis of the body and its z-axis along a line between 

the body CG and the axis of rotation of the helix. The transverse aerodynamic moments in the 

nonrolling frame can be transformed to the coordinate frame where the steady flow field exists, 

using the following relations: 

q = Helical Motion 

( ) i t

m n m n C e ~ C i ~ C iC − Ω + = + , (15) 

α& = 0 Helical Motion

( ) i t cosβ

m n m n C e ~ C i ~ C iC − Ω + = + . (16) 

Here, the transverse moments, Cm and Cn, refer to the moments about the y- and z-axes, 

respectively. 

Using this transformation, the transverse moments can be shown to have the following form for 

each of the two types of helical motion. 

q = Helical Motion 

V

R iC

V

R

V

C iC C o m o m n m

Ω

+

Ω 



 



 Ω

+ = − α α

l

& , (17) 
 11

α& = 0 Helical Motion

V

R

V

C iC C o

m n mq

Ω 



 



 Ω

+ = l . (18) 

The resulting expressions for the transverse moments are independent of time, indicating that the 

flow field is potentially steady as well. 

Similar expressions for the individual damping force coefficients can be developed using the 

same approach as applied for the moment coefficients. 

q = Helical Motion 

V

R iC

V

R

V

C iC iC o

N

o

Y Z N

Ω

+

Ω 



 



 Ω

+ = α α

l

& , (19) 

α& = 0 Helical Motion

V

R

V

C iC iC o

Y Z Nq

Ω 



 



 Ω

+ = − l . (20) 

2.4 CG Translation Relations 

If the aerodynamic coefficients have been established for a baseline configuration, it is possible 

to determine the aerodynamic coefficients for the identical configuration which has a different 

axial CG location using the CG translation relations (12). The CG translation relations for the 

damping coefficients are shown in equations 21–24. In the context of the current study, the CG 

translation relations can be used to further validate the results by comparing the theoretical 

variation of the damping coefficients with CG location with the CFD predictions. 

Nα& Nα& Cˆ = C , (21) 

Nq Nq cg Nα Cˆ = C + s C , (22) 

mα& mα& cg Nα& Cˆ = C + s C , (23) 

and 

α Nα

2 Cˆ mq = Cmq − scg (CNq − Cm ) − s cgC . (24) 

3. Computational Approach 

Computation of the viscous flow field about the flight body was accomplished by solving the 

thin-layer Navier-Stokes equations using a parabolized Navier-Stokes (PNS) technique. Because 
 12

the computations are performed in a noninertial rotating coordinate frame, the governing 

equations have been modified to include the body force terms that result from the Coriolis and 

centripedal accelerations in the rotating coordinate frame. The fluid flow relative to the rotating 

coordinate frame does not vary with time, allowing the steady (nontime varying) Navier-Stokes 

equations to be applied. The solution of the steady Navier-Stokes equations can be performed at 

a reasonable computational cost. The steady thin-layer Navier-Stokes equations are shown as 

follows: 

















+

∂

∂

+ + = ∂

∂

+

∂

∂

+

∂

∂ c Sc

ˆ S

ˆ

Re

1 Hˆ Hˆ Gˆ Eˆ Fˆ

ξ η ζ ζ . (25) 

Here, Eˆ , Fˆ , and Gˆ are the inviscid flux vectors, S

ˆ is the viscous flux vector, Hc ˆ and Scˆ are 

inviscid and viscous source terms due to the cylindrical coordinate formulation, and Hˆ is the 

source term containing the Coriolis and centrifugal force terms that result from the rotating 

coordinate frame. Each of these matrices is a function of the dependent variables represented by 

the vector ) q ( , u, v, w,e T = ρ ρ ρ ρ , where ρ and e are the density and the total energy per unit 

volume, and u, v, and w, are the velocity components in axial, circumferential, and normal 

directions. The flux terms are shown as follows: 

( ) 































+

+

=

e p U

wU

vU

uU p

U

J

1 Eˆ x

ρ

ρ

ρ ξ

ρ

( ) 































+

+

+

+

=

e p V

wV p

vV p / r

uV p

V

J

1 Fˆ

r

x

ρ η

ρ η

ρ η

ρ

φ

( ) 































+

+

+

+

=

e p W

wW p

vW p / r

uW p

W

J

1 Gˆ

r

x

ρ ζ

ρ ζ

ρ ζ

ρ

φ 

































+ +

=

x r

r

x

u f v f w f

f

f

f

0

J

1 Hˆ

ρ ρ ρ

ρ

ρ

ρ

φ

φ , (26) 

where 

x r

x r

x

W u v / r w

V u v / r w

U u

ζ ζ ζ

η η η

ξ

φ

φ

= + +

= + +

=

, (27)

and
 13

( )

( )

( )

J(x )

J r r J( x r )

J(x r ) J x

1/ x J r r

r

x

r

x x

ζ η

ξ η ξ η φ ξ η

φ ξ ζ ξ ζ

ξ ξ ζ ξ ζ

ζ φ

ζ φ φ ζ

η η φ

ξ η φ φ

=

= − = −

= = −

= = −

 )] J 1/[x ( r r = ξ φη ζ −φζ η . (28) 

The Coriolis and centrifugal acceleration terms, due to the rotating coordinate system, which are 

contained in the source term, Hˆ , are shown as follows: 

f 2 u ( R) r r r r r r = Ω× + Ω× Ω× . (29) 

The Coriolis acceleration is a function of the angular velocity of the coordinate frame with 

respect to the inertial frame, Ω

r

, and the fluid velocity vector, u,

r which can be represented by the 

velocity components, u, v, and w. The centripedal acceleration is a function of the angular 

velocity of the rotating frame, Ω

r

, and the displacement vector, R,

r between the axis of rotation 

and the local position in the flow field. The acceleration vector, f,

r

 can be written in terms of its 

components along the x, φ and r axes, x f , φ f , and r f . 

The angular velocities for the three different motions considered in this study are written in the 

following. The angular velocities are nondimensionalized in a manner consistent with the 

Navier-Stokes equations. 

Looping Motion 

r sin i

a

D

cos i

a

r D r r φ φ φ 













 Ω − 













 Ω Ω =

∞ ∞

, (30) 

α& = 0 Helical Motion 

xi

a

r D r















 Ω Ω =

∞

, (31) 

q = Helical Motion 

x r sin sin i

a

D sin cos i

a

D

cos i

a

r D r r r β β φ φ β φ 













 Ω − 













 Ω

+ 













 Ω Ω =

∞ ∞ ∞

. (32) 

The pressure, p, can be related to the dependent variables by applying the ideal gas law: 

( )





 



 = − − + + 2 2 2

u v w

2 p ( 1) e ρ γ . (33) 
 14

The turbulent viscosity, μt

, which appears in the viscous matrices, was computed using the 

Baldwin-Lomax turbulence model (13). 

The thin-layer equations are solved using the PNS technique of Schiff and Steger (14). 

Following the approach of Schiff and Steger, the governing equations, which have been modified 

here to include the Coriolis and centrifugal force terms, are solved using a conservative, 

approximately factored, implicit finite-difference numerical algorithm as formulated by Beam 

and Warming (15). Details of the implementation of the source term that contains the Coriolis 

and centrifugal force terms are given in reference (5). 

The computations presented here were performed using a shock-fitting procedure (16). This 

procedure solves the five Rankine-Hugoniot jump conditions, two geometric shock propagation 

conditions, and one compatibility equation to determine the values of the five dependent 

variables immediately behind the shock, as well as the position of the shock. By including the 

implicit part of the source term due to the rotating coordinate frame in the circumferential 

inversion, the shock-fitting procedure can be used without modification, as long as the freestream conditions are modified to account for the rotating coordinate frame. 

At the body surface, no-slip, constant wall temperature boundary conditions were applied. For 

the helical motions, the tangential velocity at the body surface was set equal to the local velocity 

of the body surface due to the solid-body rotation produced by an angular velocity equal and 

opposite to the longitudinal component of angular velocity shown in equations 31 and 32. This 

is required to produce a zero spin rate p in the nonrolling coordinate frame. 

An initial solution for the PNS marching procedure was obtained using a conical step-back 

procedure at a location of 0.2 body diameters from the nosetip. Although the perturbations to the 

flow field from the helical motions are not compatible with the conical flow assumption, the 

effect on the solution appears to be small. Moving the starting plane to 0.1 body diameters from 

the nose tip resulted in <0.5% variation in the computed damping moments. 

The computational results presented here were obtained using a grid that consisted of 60 points 

between the body and the shock. In the circumferential direction, gridding was performed over a 

360° sector because of the lack of symmetry from the combination of angle of attack, spin, and 

coning or helical motion. Thirty-six grid points were used in the circumferential direction. In 

longitudinal direction, 78 marching steps were utilized for each body diameter of length. To 

ensure adequate grid resolution within the boundary layer, the grid spacing at the body surface 

was adapted to maintain nondimensional boundary layer coordinate y+

 between 2 and 3 in 

accordance with previously published results (17). Grid resolution studies for the 9-caliber body 

(middle CG location) showed <2% variation in the computed pitch-damping coefficient sum 

when the grid resolution was decreased by 25% in each of the three coordinate directions. 

Similarly, when the grid in the circumferential and marching directions was doubled and the grid 

in the radial direction was increased by 50%, the computed pitch-damping coefficient sum varied 
 15

by only 1%. Differences in the damping coefficient mq

C on the coarse and fine mesh, relative to 

the baseline grid, were <1%. The damping coefficient mα& C showed a larger variability on a 

percentage basis on the coarse and fine grid relative to the baseline grid (6% and 3%, 

respectively), although absolute differences between the predicted values of the coefficient were 

similar to the differences between the predicted pitch-damping coefficient sum. The smaller 

magnitude of Cm , α& relative to the pitch-damping coefficient sum, contributed to a larger 

percentage error. The computations for the baseline grid, which were performed using a Cray YMP supercomputer, typically required <10 min of CPU time for complete calculation over a 

single configuration. 

4. Results 

Computations of the individual coefficients that comprise the pitch-damping force and moment 

coefficient sums were performed using helical and looping motions. The computations were 

performed for the ANSR series of bodies shown in figure 1. Results for length-to-diameter 

(L/D) ratio = 5, 7, and 9 bodies are presented here for flight velocities of Mach 1.8 and 2.5. 

Three different CG locations were considered for each body length, as shown in table 1. 

Aerodynamic range tests of the ANSR (18) were used to benchmark the predicted pitch-damping 

coefficient sum. 

Table 1. CG location for ANSR flight bodies. 

 Longitudinal CG 

(Calibers From Nose) 

L/D Forward Middle Rearward 

5 2.5 3.0 3.5 

7 3.251 4.037 4.812 

9 4.0 5.038 5.885 

The effect of angular rate and the rotational velocity ratio for the q = 0 helical motion and α& = 0

helical motion was examined for the L/D = 9 ANSR body (forward CG) at a flight velocity of 

Mach 2.5. The force and moment expansion for both of these motions indicates a linear 

variation of the forces and moments is expected with angular rate and rotational velocity ratio 

within the regime where linear aerodynamic theory is expected to be valid. The predictions were 

performed for the longest ANSR body because the longer bodies typically exhibit more 

nonlinear aerodynamic behavior. Thus, the onset of nonlinear behavior of the aerodynamics 

with angular rate or angle of attack would be more evident, and the limits of linear aerodynamic 

behavior could be more readily identified. 
 16

The effect of angular rate for the q = 0 motion was examined by performing computations at 

several angular velocities but at a fixed total angle of attack of 2°. This means that as the angular 

velocity was increased, the radius Ro was decreased so that the product R (and the total angle oΩ

of attack) was constant. Figure 8 shows the variation of the aerodynamic moment Cm with 

angular velocity, and the variation is seen to be linear across the range of angular velocities of 

interest here. 

Figure 8. Variation of aerodynamic moment Cm with angular velocity, q = 0 helical motion, 

M = 2.5, ANSR, sin–1 δ = 2°. 

For the q = 0 helical motion, the sine of the angle of attack, δ , varies directly with the rotational 

velocity ratio, V

RoΩ (see equation 8). Thus, as the rotational velocity ratio increases toward 1, 

the angle of attack of the vehicle also increases for the q = 0 helical motion. (In the limit as 

1

V

Ro = Ω , for q = 0 helical motion, the CG performs a circular looping motion with the body at 

90° angle of attack.) The desire here is to define the limits where the forces and moments are 

expected to vary linearly with δ or 

V

RoΩ . Computations of q = 0 helical motion were 

performed with the angular rate held fixed ( 0.01

V

D = Ω ), while the helix radius was varied to 

produce the desired angle of attack. The aerodynamic moment Cm, shown in figure 9, shows a 
 17

Figure 9. Variation of aerodynamic moment Cm with sine of the angle of attack, q = 0 helical 

motion, M = 2.5, ANSR, 

V

ΩD . 

linear variation with δ up to angles of attack of ~7°. This behavior is similar to that observed 

for coning motion (5). 

For the α& = 0 helical motion, the effect of angular rate was examined for rotational velocity 

ratio, sin 0.0349

V

Ro = = Ω β . Again, as the angular velocity was increased, the radius Ro was 

decreased so that the product R (and oΩ sin ) was constant. The variation of the aerodynamic β

moment Cm with angular velocity, shown in figure 10, is seen to be linear across the range of 

angular velocities of interest here. 

The effect of the rotational velocity ratio sin β V

Ro = Ω on the forces and moments for the α& = 0

helical motion was also examined, and results are shown in figure 11. These predictions were 

made for a constant angular velocity, ( 0.01

V

D = Ω ). The aerodynamic moment Cm is seen to be 

linear with rotational velocity ratios from 0 to 1. At first glance, a nonlinear variation might be 

expected since nonlinear variations with δ were seen for the q = 0 helical motion. It is noted 
 18

Figure 10. Variation of aerodynamic moment Cm with angular velocity, α& = 0 helical motion, 

M = 2.5, ANSR, β = 2°. 

Figure 11. Variation of aerodynamic moment Cm with rotational velocity ratio, α& = 0 helical 

motion, M = 2.5, ANSR, 0.010

V

D = Ω . 
 19

that the parameter sin β V

Ro = Ω is not associated with the angle of attack for this motion 

because the α& = 0 helical motion is performed at 0° angle of attack. The parameter, , sin β is 

simply the rotational velocity ratio which for the α& = 0 helical motion reflects the aspect ratio of 

the helix. The limiting case of a rotational velocity ratio of 1 corresponds to a circular looping 

motion discussed previously. 

Using these results as a guide, predictions of the pitch-damping coefficients were made for each 

of the three body lengths and three CGs at Mach 1.8 and 2.5. Predictions were made at an 

angular velocity of 0.01

V

D = Ω and a rotational velocity ratio, 0.0349, VRo = Ω where the linear 

aerodynamic theory has been shown to be valid. 

Figures 12–14 show predictions of the pitch-damping moment coefficients, mqC and mα& C , as a 

function of the CG position for the L/D = 5, 7, and 9 bodies at Mach 2.5. Predictions obtained 

using the PNS computational approach are shown, along with results obtained with slender body 

theory. Both PNS and slender body predictions show that the pitch-damping coefficient mq

C is 

larger than mα& C for all the CG positions examined. The PNS prediction of the damping 

coefficient, mα& C , is nearly zero at the rearward CG for each of the body lengths. The trends 

shown by the slender body results are similar to the PNS results, although the slender body 

predictions are generally lower in magnitude. The PNS predictions of the variation of mq

C with 

CG position shown in figures 12–14 agree with the theoretical variation (equation 24) to within 

0.25% across the range of CG positions examined. The PNS predictions of the variation of mα& C

are essentially identical to the theoretical variation (equation 23) because the governing 

equations, boundary conditions, and flow field are independent of the axial CG location. 

Figures 15–17 show similar predictions of the pitch-damping force coefficients, Nq

C and Nα& C , 

as a function of CG position for the L/D = 5, 7, and 9 bodies at Mach 2.5. The PNS predictions 

of the pitch-damping force coefficients show a similar trend to the predictions made with slender 

body theory, though the slender body theory results are larger in magnitude. The agreement does 

seem to improve slightly as the body length increases. 

Because ballistic bodies traverse a nearly rectilinear path, it is impossible to extract the 

individual components of the damping coefficients from the experimental data. Thus, no 

comparison of the predicted individual coefficients could be made with experimental data. 

However, the individual coefficients can be summed to produce the pitch-damping coefficient 

sum, and the results compared with the coning motion results and with experiment. Figures 

18–20 show comparisons of the pitch-damping coefficient sum (obtained by summing the 

individual coefficients) with experimental data and slender body theory results. The results 
 20

Figure 12. Variation of damping moment coefficients with CG location, M = 2.5, ANSR, 

L/D = 5. 

Figure 13. Variation of damping moment coefficients with CG location, M = 2.5, ANSR, 

L/D = 7. 
 21

Figure 14. Variation of damping moment coefficients with CG location, M = 2.5, ANSR, 

L/D = 9. 

Figure 15. Variation of damping force coefficients with CG location, M = 2.5, ANSR, 

L/D = 5. 
 22

Figure 16. Variation of damping force coefficients with CG location, M = 2.5, ANSR, 

L/D = 7. 

Figure 17. Variation of damping force coefficients with CG location, M = 2.5, ANSR, 

L/D = 9. 
 23

Figure 18. Variation of damping moment coefficient sum with CG location, M = 2.5, ANSR, 

L/D = 5. 

Figure 19. Variation of damping moment coefficient sum with CG location, M = 2.5, ANSR, 

L/D = 7. 
 24

Figure 20. Variation of damping moment coefficient sum with CG location, M = 2.5, ANSR, 

L/D = 9. 

obtained by summing the predicted coefficients, mqC and Cm , α& are within 1.0% or less of the 

pitch-damping moment coefficient sum predicted using coning motion. 

To provide additional validation of the current approach, the predictions were also compared 

with the nonlinear unsteady potential equation results obtained by Devan (19) for the L/D = 5 

ANSR configuration. The variation in the two damping coefficients with Mach number is shown 

in figure 21. In general, the results show modest variations in the predicted coefficients across 

the Mach number range except for the slender body predictions which are independent of Mach 

number. The nonlinear potential solution of Devan shows good agreement with the PNS 

predictions with a similar variation with Mach number predicted. (It is noted that Devan did not 

explicitly determine the damping coefficient mq C , but instead computed mα& C and 

(C Cm ) mq α& + from which mq C can be obtained.) The approach of Devan overpredicts the two 

damping coefficients by 7% for mq

C and ~25% for Cm , α& compared with the PNS results. The 

predicted value of mq C from slender body theory differs by as much as 25% with the PNS 

predictions. The slender body theory predicts a nearly zero value for the damping coefficient 

Cm , α& while the PNS predictions show this coefficient is ~ 25%–40% of the value of mqC at the 

middle CG location. 
 25

Figure 21. Variation of the pitch-damping moment coefficients with Mach number, ANSR, 

L/D = 5, middle CG location. 

A comparison of the Mach number variation of the pitch-damping coefficient sum for the various 

approaches with aerodynamic range data is shown in figure 22. Compared with the slender body 

theory and the nonlinear unsteady potential equation results of Devan, the PNS results show the 

best agreement with the experimental data. The approach of Devan shows generally a similar 

variation with Mach number as the PNS approach and the experimental data, but appears to show 

a slight overprediction (~10%) across the Mach number range. This overprediction is consistent 

with the overprediction seen previously for the individual coefficients. The slender body theory 

results, which exhibit no dependence on Mach number, consistently underpredicts the pitchdamping sum by >30%. 

The distribution of the pitch-damping force and moment coefficients over the ANSR L/D = 5 

body is shown in figures 23 and 24. The force coefficient Nα& C is positive along the body length 

with most of the force being generated at the aft end of the body. As a result, the moment 

coefficient mα& C also shows its largest contributions from the aft portion of the body. On the 

other hand, for the force coefficient, Nq C , both the nose and cylindrical afterbody produce 

contributions to the force coefficient which are similar in magnitude but opposite in sign. This 

essentially imposes a couple on the body whose magnitude is reflected in the moment 

coefficient,C . mq
 26

Figure 22. Variation of the pitch-damping moment coefficient sum with Mach number, 

ANSR, L/D = 5, middle CG location. 

Figure 23. Development of pitch-damping force coefficients over ANSR body, M = 2.5, 

L/D = 5, middle CG position. 
 27

Figure 24. Development of pitch-damping moment coefficients over ANSR body, M = 2.5, 

L/D = 5, middle CG position. 

5. Conclusion 

A method for determining the individual force and moment coefficients that comprise the pitchdamping force and moment coefficient sum has been presented, along with sample results for a 

family of axisymmetric projectile geometries. The sum of the two individual coefficients is 

identical to previous predictions of the pitch-damping coefficient sums obtained using coning 

motion and is in excellent agreement with experimental data. Additional validation of the 

approach is obtained through comparisons with prior results from numerical solutions of the 

nonlinear unsteady potential equation. The individual coefficients show qualitative agreement 

with results obtained using the more approximate slender body theory. The method provides an 

efficient means of determining the individual coefficients which may be quite difficult to obtain 

using experimental means. 
 28

6. References 

1. Schiff, L. B. Nonlinear Aerodynamics of Bodies in Coning Motion. AIAA Journal 1972, 10

(11), 1517–1522. 

2. Agarwal, R.; Rakich, J. V. Computation of Supersonic Laminar Viscous Flow Past a 

Pointed Cone at Angle of Attack in Spinning and Coning Motion. AIAA Paper, July 1978; 

78–1211. 

3. Lin, T. C. A Numerical Study of the Aerodynamics of a Reentry Vehicle in Steady Coning 

Motion. AIAA Paper, August 1978; 78–1358. 

4. Weinacht, P.; Sturek, W. B. Navier-Stokes Predictions of Pitch-Damping for Finned 

Projectiles Using Steady Coning Motion. Proceedings of the AIAA 8th Applied 

Aerodynamics Conference, AIAA, Washington, DC, August 1990; AIAA Paper 90-3088, 

632–642. 

5. Weinacht, P.; Sturek, W. B.; Schiff, L. B. Navier-Stokes Predictions of Pitch-Damping for 

Axisymmetric Shell Using Steady Coning Motion. Proceedings of the 1991 Atmospheric 

Flight Mechanics Conference, AIAA, Washington, DC, August 1991; AIAA Paper 91-2855, 

89–100. 

6. Weinacht, P.; Sturek, W. B.; Schiff, L. B. Navier-Stokes Predictions of Pitch-Damping for 

Axisymmetric Shell Using Steady Coning Motion. Journal of Spacecraft and Rockets 1997,

34 (6), 753–761. 

7. Weinacht, P. Navier-Stokes Predictions of Pitch-Damping for a Family of Flared Projectiles. 

Proceedings of the 9th Applied Aerodynamics Conference, AIAA, Washington, DC, 

September 1991; AIAA Paper 91-3339, Vol. 2, 1025–1034. 

8. Weinacht, P. Navier-Stokes Predictions of the Individual Components of the Pitch-Damping 

Coefficient Sum. Proceedings of the 1995 Atmospheric Flight Mechanics Conference, 

AIAA, Washington, DC, August 1995; AIAA Paper 95-3458, 291–301. 

9. Weinacht, P. Navier-Stokes Predictions of the Individual Components of the Pitch-Damping 

Coefficient Sum. Journal of Spacecraft and Rockets 1998, 35 (5), 598–605. 

10. Weinacht, P. Prediction of Pitch-Damping Aerodynamic Derivatives Using Navier-Stokes 

Computational Techniques. Ph.D. Thesis, University of Delaware, Newark, DE, 1996. 

11. Qin, N.; Ludlow, D. K.; Shaw, S. T.; Edwards, J. A.; Dupuis, A. Calculation of Pitch 

Damping Coefficients for Projectiles. AIAA Paper 97-0405, January 1997. 
 29

12. Murphy, C. H. Free Flight Motion of Symmetric Missiles; BRL-TR-1216; U.S. Army 

Ballistic Research Laboratory: Aberdeen Proving Ground, MD, 1963. 

13. Baldwin, B. S.; Lomax, H. Thin Layer Approximation and Algebraic Model for Separated 

Turbulent Flows. AIAA Paper 78-257, January 1978. 

14. Schiff, L. B.; Steger, J. L. Numerical Simulation of Steady Supersonic Viscous Flow. AIAA 

Journal 1980, 18 (12), 1421–1430. 

15. Beam, R.; Warming, R. F. An Implicit Factored Scheme for the Compressible NavierStokes Equations. AIAA Journal 1978, 16 (4), 393–402. 

16. Rai, M. M.; Chaussee, D. S. New Implicit Boundary Procedure: Theory and Applications. 

AIAA Paper 83-0123, January 1983. 

17. Sturek, W. B.; Schiff, L. B. Computations of the Magnus Effect for Slender Bodies in 

Supersonic Flight. Proceedings of the AIAA Atmospheric Flight Mechanics Conference, 

AIAA, New York, August 1980; AIAA Paper 80-1586-CP, 260–280. 

18. Murphy, C. H.; Schmidt, L. E. The Effect of Length on the Aerodynamics Characteristics of 

Bodies of Revolution in Supersonic Flight; BRL-TR-876; U.S. Army Ballistic Research 

Laboratory: Aberdeen Proving Ground, MD, 1953. 

19. Devan, L. Nonaxisymmetric Body, Supersonic, Inviscid Dynamic Derivative Predictions; 

NSWC TR 89-99; U.S. Naval Surface Warfare Center: Dahlgren, VA, 1989. 
 30

INTENTIONALLY LEFT BLANK. 
 31

List of Abbreviations and Symbols 

∞a Freestream speed of sound 

Cm Pitching moment coefficient, 

V S D

2

1

M

ref

2 ρ

mα C Pitching moment coefficient slope with respect to angle of attack, ∂α

∂Cm

mα& C Pitch-damping moment coefficient slope, 





 





∂

∂

V

D

Cm

α&

mq C Pitch-damping moment coefficient slope, 





 





∂

∂

V

q D

Cm

mq mα& C + C Pitch-damping moment coefficient 

Cn Side moment coefficient 

Cn pα Magnus moment coefficient, 

∂α



 





∂

∂

V

p D

Cn

2

C Normal force coefficient, N

ref

2 V S

2

1

F

ρ

Nα C Normal force coefficient slope with respect to angle of attack, ∂α

∂CN

Nα& C Pitch-damping force coefficient slope, 





 





∂

∂

V

D

CN

α&

Nq

C Pitch-damping force slope, 





 





∂

∂

V

q D

CN

Nq Nα& C + C Pitch-damping force coefficient 
 32

D Reference diameter 

e Total energy per unit volume 

Gˆ F, ˆ E, ˆ Flux vectors in transformed coordinates 

x r f ,f ,f φ Component of acceleration vector fr

 in the x , φ , and r directions 

F Force 

Hˆ Source term in Navier-Stokes equations due rotating coordinate frame 

Hc ˆ Inviscid source term due to cylindrical coordinate formulation 

J Jacobian 

l Characteristic length, typically D 

L Body length 

M Moment 

M Freestream Mach number 

p Pressure, as used in Navier-Stokes equations 

p Spin rate, as used in roll equations 

q,r Transverse angular rates of body along y and z directions 

r Radial coordinate as used in Navier-Stokes equations 

R

r

 Displacement vector between the axis of rotation and location in flow field 

R Helix radius or radius of o circular arc for looping motion 

Re Reynolds number, ∞ ρ ∞ μ ∞ a D /

s Distance downrange 

cg s Center of gravity shift in calibers 

S Reference area, ref 4

D S

2

ref

π

=

S

ˆ Viscous flux vector 

Sc

ˆ Additional viscous terms due to cylindrical coordinate formulation 

t Time 
 33

u Velocity components in , v,w the three directions 

U Vehicle velocity alone helix axis 

V Magnitude of freestream velocity 

V∞

r Freestream velocity vector 

x Axial, horizontal, a , y, z nd vertical coordinates 

e e e x Three-dimensional coordina , y , z te in earth-fixed system 

x Axial location of center of gravity cg

+ y Nondimensional boundary layer coordinate 

Greek Symbols 

α Angle of attack 

α& Angular rate associated with angle of attack 

β Angle between helix axis and longitudinal body axis for α& = 0 helical motion 

γ Ratio of specific heats, in Navier-Stokes equations 

γ Cosine of total angle of attack, as used in the aerodynamic force and moment 

equations 

δ Sine of the total angle of attack 

θ Angular orientation of body relative to inertial frame for looping motion 

t μ,μ Laminar and turbulent viscosity 

μ

~ Complex transverse angular velocity 

ξ,η,ζ Transformed coordinates in the Navier-Stokes equations 

ξ

~ Complex angle of attack 

ρ Density 

φ Circumferential coordinate, measured from the vertical axis 

Ω Angular rate associated with looping and helical motions 
 34

Superscripts 

( )& Rate of change with respect to time 

( )′

 Rate of change with respect to space 

( ) ~ Referenced to nonrolling coordinate frame 

Subscripts 

∞ Quantity evaluated at freestream conditions 
NO. OF NO. OF 

COPIES ORGANIZATION COPIES ORGANIZATION

35

 1 DEFENSE TECHNICAL 

 (PDF INFORMATION CENTER 

 Only) DTIC OCA 

 8725 JOHN J KINGMAN RD 

 STE 0944 

 FT BELVOIR VA 22060-6218 

 1 COMMANDING GENERAL 

 US ARMY MATERIEL CMD 

 AMCRDA TF 

 5001 EISENHOWER AVE 

 ALEXANDRIA VA 22333-0001 

 1 INST FOR ADVNCD TCHNLGY 

 THE UNIV OF TEXAS 

 AT AUSTIN 

 3925 W BRAKER LN STE 400 

 AUSTIN TX 78759-5316 

 1 US MILITARY ACADEMY 

 MATH SCI CTR EXCELLENCE 

 MADN MATH 

 THAYER HALL 

 WEST POINT NY 10996-1786 

 1 DIRECTOR 

 US ARMY RESEARCH LAB 

 AMSRD ARL D 

 DR D SMITH 

 2800 POWDER MILL RD 

 ADELPHI MD 20783-1197 

 1 DIRECTOR 

 US ARMY RESEARCH LAB 

 AMSRD ARL CS IS R 

 2800 POWDER MILL RD 

 ADELPHI MD 20783-1197 

 3 DIRECTOR 

 US ARMY RESEARCH LAB 

 AMSRD ARL CI OK TL 

 2800 POWDER MILL RD 

 ADELPHI MD 20783-1197 

 3 DIRECTOR 

 US ARMY RESEARCH LAB 

 AMSRD ARL CS IS T 

 2800 POWDER MILL RD 

 ADELPHI MD 20783-1197 

ABERDEEN PROVING GROUND

 1 DIR USARL 

 AMSRD ARL CI OK TP (BLDG 4600) 
NO. OF 

COPIES ORGANIZATION 

36

 1 ARROW TECH ASSOCIATES 

 W HATHAWAY 

 1233 SHELBURNE RD 

 D 8 

 BURLINGTON VT 05403 

 1 OREGON STATE UNIVERSITY 

 DEPT OF MECH ENGINEERING 

 M COSTELLO 

 CORVALLIS OR 97331 

 1 ATK 

 R DOHRN 

 MN07 LW54 

 4700 NATHAN LN N 

 PLYMOUTH MN 55442 

 1 AEROPREDICTION INC 

 F MOORE 

 9449 GROVER DR 

 SUITE 201 

 KING GEORGE VA 22485 

 1 COMMANDER ARDEC 

 AMSTR AR FSF X 

 W TOLEDO 

 BLDG 95 SOUTH 

 PICATINNY ARSENAL NJ 07806-5000 

ABERDEEN PROVING GROUND

 21 DIR USARL 

 AMSRD ARL WM B 

 A HORST 

 AMSRD ARL WM BA 

 D LYON 

 AMSRD ARL WM BC 

 M BUNDY 

 G COOPER 

 J DESPIRITO 

 J GARNER 

 B GUIDOS 

 K HEAVEY 

 J NEWILL 

 P PLOSTINS 

 J SAHU 

 S SILTON 

 D WEBB 

 P WEINACHT (5 CPS) 

 A ZIELINSKI 

 AMSRD ARL WM BD 

 B FORCH 

 AMSRD ARL WM BF 

 S WILKERSON