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AIAA-2003-3670Cartesian-Grid Simulations of aCanard-Controlled Missile with a Spinning Tail

Scott M. MurmanELORETMoffett Field, CA

Michael J. AftosmisNASA Ames Research CenterMoffett Field, CA

21st AIAA Applied Aerodynamics ConferenceJune 23-26, 2003 / Orlando, FL

AIAA-2003-3670

Cartesian-Grid Simulations of a

Canard-Controlled Missile with a Spinning Tail

Scott M. Murman∗

ELORETMS T27B

Moffett Field, CA 94035smurman@nas.nasa.gov

Michael J. Aftosmis†

NASA Ames Research CenterMS T27B

Moffett Field, CA 94035aftosmis@nas.nasa.gov

Abstract

The paper presents a series of simulations of a geometrically-complex, canard-controlled, supersonicmissile with free-spinning tail fins. Time-dependent, relative-motion simulations were performed usingan inviscid Cartesian-grid-based method at three angles of attack. Two methods of modeling thespinning tail section are compared; an imposed tail spin rate such that the net rolling moment on theempennage is zero, and a free-to-spin configuration using a constrained 6-DOF motion. Computationalresults are compared with high-resolution Navier-Stokes computations. The results indicate that thechoice of a static, forced-spin, or free-to-spin analysis cannot in general be made a priori. Further, thebehavior of the dynamic tail section is likely multi-valued, and hence the state for any configuration isdependent upon the past history of the missile.

1 Introduction

Over the past decade, static ComputationalFluid Dynamics (CFD) simulations over increas-ingly complex vehicles have become commonplace.In this evolution, non-body-fitted Cartesian gridmethods have proven to be particularly usefulfor automatically meshing geometrically-complexvehicles[1–6]. Recently this class of meshing andsolution techniques has been extended to dynamicsimulations[7–10], where components of the ge-ometry move in some manner during the simu-lation. This makes highly-automated simulationsof complex three-dimensional vehicles with compo-nents in relative motion more feasible. The currentwork adopts a non-body-fitted Cartesian methodto study the performance of a supersonic, canard-controlled missile with a free-spinning tail. On thistype of vehicle, the tail fins are free to spin as a unit

∗Senior Research Scientist, Member AIAA†Research Scientist, Senior Member AIAACopyright c©2003 by the American Institute of Aero-

nautics and Astronautics, Inc. The U. S. Government has aroyalty-free license to exercise all rights under the copyrightclaimed herein for Governmental purposes. All other rightsare reserved by the copyright owner.

around the missile longitudinal axis. As a result,torque from aerodynamic loads on the empennagecauses the fin system to spin, even under steady-state flight conditions.

Missiles with dynamic components can pose sig-nificant challenges for numerical simulation. In thepresent example, the spinning tail is a by-productof the forces on the missile and is integral to itsaerodynamic performance. Nevertheless, the per-formance of the missile is often characterized by thespin-averaged aerodynamic coefficients, and hencetime-dependent, moving-body simulations are re-quired to predict even static stability and control(S&C) information.∗ Moreover, the spin-rate of thefin system is governed not only by the wind vectorand canard deflections, but also by the strength andlocation of the convected canard vortices, whose in-duced velocity field differentially loads the tail finsat low angles of attack. The need to convect thecanard vortices over the length of the missile tointeract with the fin system directly impacts thesize of the computational mesh required for ac-

∗Static here means the absence of higher-order dynamicstability derivatives, but does still include spinning tail sec-tion.

1

curate numerical simulation. The combination ofthese factors makes for CPU-intensive simulationssince the physics requires both highly-resolved spa-tial grids and time-dependent, moving-body solu-tion methodologies.

The paper details the missile geometry underconsideration, important features of the computa-tional mesh, and the numerical method used forthe simulations. The numerical investigations firstconcentrate on the flowfield at α = 4.0◦, wherethe canard/tail interactions are strongest, and thespin rate of the tail is expected to be highest.Steady-state simulations of the missile with thefins fixed at various azimuths around the missileaxis establish a zero-spin-rate baseline. Dynamicsimulations are then performed with an imposedspin rate on the tail. An iterative process is usedto determine the spin rate which predicts a zerospin-averaged torque on the tail. These fixed spinrate simulations are compared with free-to-spinsimulations obtained using a coupled CFD/6-DOFapproach. The computational results are com-pared with highly-resolved Navier-Stokes numericalsimulations[11]. The final section extends the anal-ysis by considering the variation of tail behaviorwith angle of attack for fixed canard settings. Thestatic, forced-spin, and free-to-spin analyses are allused to provide detailed insight into the complexdynamic behavior of the tail.

2 Numerical Method

2.1 Geometry and ComputationalMesh

Figure 1 shows three views of the canard-controlled missile used for this study. The missileis depicted with the tail in the + position, corre-sponding to a tail rotation angle of φtail = 45◦.Zero rotation angle is obtained when the tail isin the × position, and positive rotation is clock-wise when observed from the missile nose. Thecanards are shown in their deflected position –pitched asymmetrically with δc = 16◦ to commanda starboard yaw of the missile. The missile bodyhas a cylindrical cross-section with a fineness ratio(length/diameter ratio) of about 15.0. Two con-duits, which are raised off the body and anchored atregular intervals leaving a small gap, run the lengthof the body. In addition, the missile has a ring of10 bluff protuberances at roughly the mid-station

of the body, and several others at the aft end nearthe free-spinning tail fins. The leading edges of thefins extend forward along the body, and the rootstation is cutaway to clear both the conduits andother hardware as the fins sweep over the missilebody. The minimum clearance is about 1/8th ofthe fin thickness, and occurs when the fin passesover the protuberances on the aft missile body.

Detailed enlargements of the surface triangula-tion near the nose, mid-body and tail fins are shownin Fig. 2. The tail is in the + position which putsthe upper and lower fins over two of the conduitsand the other two fins over protuberances in theaft missile forebody. This triangulation was pro-duced directly from CAD solids without requiringany user input using the automatic triangulationsoftware described in [12]. This software uses theCAPRI library[13, 14] to access the CAD geometryusing the CAD system’s native query routines andgeometry engine. The final triangulation (shown)uses approximately 400,000 triangles which wereused as input to the Cartesian mesh generationsystem[5].

Tail Rotation

Canards Set to ProvideStarboard Yaw

cδ

Figure 1: Front, side and isometric view of generic missilewith free-spinning tail. For the simulations presented, all 4canards are deflected δc = 16◦ to command a starboard yawof the missile. The tail is shown in the + configuration whichcorresponds to a tail rotation angle, φtail = 45◦. φtail = 0◦

is obtained with the empennage in the × position.

(a) (b) (c)

Figure 2: Surface mesh detail of missile configuration,400,000 triangles.

2

Figure 3 shows the non-body-fitted baselineCartesian mesh used for the simulations. This fig-ure shows the tail fin in the × position (φtail = 0◦),and the mesh is displayed by several cutting planesbehind and perpendicular to the missile axis. Incomputing flows around canard-controlled missiles,it can be very important to avoid excessively dis-sipating the canard vortices as they convect thelength of the missile body. To provide this resolu-tion, the mesh has a pre-specified adaptation regioncovering the entire missile, and within this regionthe mesh is refined 3 levels further driven by sur-face curvature as described in [5]. In addition tothe canard vortices, a pre-specified adaptation re-gion is designed to capture the shocks generated bymany of the surface features on the missile body.Since streamlines passing through these shocks willimpact the spinning fins, resolution and propaga-tion of these shocks may be important. Resolu-tion requirements for the baseline mesh were estab-lished using guidelines from previous simulations ofcanard-controlled missiles[10], and by performing amesh resolution study with the current geometry.As the missile tail spins over the course of the simu-lation the mesh responds to track the body motion,re-adapting to the new geometry at each timestep(cf. [9, 10]). The snapshot shown here has approx-imately 4 million cells, and this total number ofcells remains roughly constant over the coarse of adynamic simulation with the tail section spinning.

Figure 3: Cutting planes through the non-body-fittedCartesian mesh used in simulations. Missile is shown withfins in the × position and canards deflected asymmetricallyδc = 16◦. 4M Cartesian cells.

2.2 Cartesian Moving-Body FlowSolver

In order to simulate a missile with a spinningtail section, a scheme that allows rigid bodies tomove relative to each other during a simulation isneeded. A general numerical scheme for solvingtime-dependent flows with (optional) rigid-bodymotion for unstructured Cartesian meshes was de-veloped from the parallel, steady-state solver de-scribed in [15]. A brief overview of the schemewill be presented here, and complete details canbe found in [9].

2.2.1 Dual-time formulation

Extension of the steady-state flow solver to time-dependent flows was accomplished using a dual-time formulation (cf. Refs. [16, 17]),

dQdτ

+ R∗ (Q) = 0

R∗ (Q) =∂Q∂t

+ R (Q)(1)

where τ is referred to here as “pseudo-time”, andis the iterative parameter, and t is the physicaltime. Q is the vector of conserved variables, andR (Q) is an appropriate numerical quadrature ofthe flux divergence, 1

V

∮Sf · ndS. As dQ

dτ → 0 thetime-dependent formulation is recovered. The par-allel multi-grid solver described in [15] is used toefficiently converge the inner pseudo-time integra-tion. This is similar to the scheme outlined byJameson[18], however, the semi-implicit approachof Melson et al.[19] is used here for the physicaltime-derivative term.

Various time-dependent schemes can be con-structed for Eqn. 1 by appropriately discretizingthe time derivative. In the current work, it’s de-sirable to use an unconditionally-stable, implicitscheme to allow a large timestep to be chosen basedupon physical considerations rather than a poten-tially smaller stability-limited timestep. In theCartesian embedded-boundary scheme, the cut-cellpolyhedra can have arbitrarily small volumes, and astability limit can be very restrictive. Using a largetimestep also reduces the amount of computationalwork required to process the moving geometry andmesh through a complete simulation. In the currentwork, the backward Euler and 2nd-order backwardtime-integration schemes have both been utilized.

3

Uncut

Swept

Cut

t = n

t = n + 1

Figure 4: Schematic of a rigid-body moving through aCartesian mesh. Cells cut by the geometry at each timestepare outlined in blue, and cells swept by the geometry overthe timestep are tinted yellow.

2.2.2 Relative motion

Figure 4 shows a schematic of a rigid-body mov-ing through a fixed Cartesian mesh over one dis-crete timestep. Cells cut at the beginning and endof the timestep are outlined in blue, and the shadedregion highlights cells which have been “swept” bythe body through the timestep. These swept cellschange volume and shape over the timestep, andcan appear or disappear (or both) as well. Awayfrom the swept region, the cells don’t change andtherefore require no special treatment. The swept-cells, however constitute the major challenge sincethe deformation of these cells over the timestepneeds to be taken into account in order to satisfythe governing equations. The equations of motionfor the deforming cells can be written in an integralconservation form as

d

dt

∫V (t)

QdV = −∮

S(t)

f · ndS S = ∂V (2)

f · n =

ρun

ρunu + pnρune + pu · n

un = (u−w) · n

Here w is the velocity of the moving boundary withrespect to the Eulerian frame.

Integrating the deforming-cell governing equa-tions for a representative cell j using the backward-

Euler scheme gives

(QV )n+1j − (QV )n

j −∑

ahead

(QV )n =

−∆t∑ (

f̂ · n∆S)n+1

j(3)

∑ahead (QV )n is a conservation correction term

which represents the flux out of the swept cells overa timestep (cf. Ref. [9]). Equation 3 can be numer-ically integrated using the dual-time scheme out-lined above. The terms (QV )n

j and∑

ahead (QV )n

become fixed source terms in the dual-time scheme.(QV )n however is only available on the mesh attime level n, while it is required on the mesh attime level n+1 in order to integrate Eqn. 3. (QV )n

is conservatively transferred from the mesh at timelevel n to the new mesh at n + 1 external to theflow solver. The transfer of the solution betweentwo volumes meshes takes advantage of the space-filling-curve ordering of the cells in the Cartesianmeshes (cf. Ref. [15, 20]). This allows the transferto be performed very efficiently, requiring only twosweeps over the mesh cell list.

3 Numerical Results

The general 3-D Cartesian scheme outlinedabove is utilized to simulate the canard-controlledmissile with spinning tail section described inSec. 2.1. The Cartesian moving-body flow solver in-cludes an interface for the GMP infrastructure[21].This interface allows the entire analysis to proceedin a completely automated manner. The analystdefines a static configuration “space” of discrete tailorientations to calculate, defines a prescribed mo-tion of the tail section for a dynamic simulation,and specifies a constrained 6-DOF simulation forthe same tail section all through an interactive GUIapplication. These specified motions and states arethen transfered to the flow solver and the static anddynamic analyses proceed in parallel without userintervention.

Since the flow conditions considered in this workare supersonic (M∞ = 1.6), the geometry upstreamof the tail section is static, and the tail sectionhas horizontal and vertical symmetry, the flowfieldwithin the tail section is periodic every 90◦ of spin.This periodicity was confirmed by the initial dy-namic simulations. As such, it’s only necessary tosimulate the motion of the tail section through 90◦

of rotation (after the initial transient). A baseline

4

angle of attack of α = 4.0◦ is used to introduce thetail forced-spin-rate and free-to-spin approaches, asthe canard/tail interactions are greatest at this an-gle of attack, leading to the highest tail spin rate.Next, the trends of the spinning tail section withangle-of-attack variation are examined by simulat-ing α = 0.0◦ and 12.0◦, both with the tail held fixedand spinning.

3.1 Static Baselines

In order to provide a baseline for comparisonwith dynamic, spinning-tail computations, a se-ries of static, steady-state simulations with the tailfixed at various (non-uniform) azimuthal orienta-tions are undertaken for each configuration exam-ined. A converged steady-state solution with thetail section in the × configuration (φtail = 0◦) isused to introduce the features of the flowfield inFig. 5. Mach number contours at cutting planesalong the longitudinal axis of the missile highlightthe convection of the canard vortices. A schematicnext to the contour planes shows the orientationand direction of spin of the vortices. The vorticesshed from the tips of the NW, NE, and SE canards(following the compass directions viewed from thenose) convect down the length of the body. Thevortex from the NE canard is stronger than the NWor SE vortices, as that canard is pitched up, whilethe other canards are pitched down. The vortexshed from the SW canard is “trapped” by the bodyas it convects upwards and dissipates. The asym-metric pitch of the canards causes an induced veloc-ity which drives the vortices into the NE quadrant,where they impact the tail section. The induced ve-locity from the canards changes the local angle ofattack seen by the tail fins, and the canard vorticesalso provide suction on the tail fins. This can beseen quantitatively in Fig. 6, where the variation oftail rolling moment with angle of rotation from the× position is shown for a series of static solutions.A negative tail rolling moment would cause the tailsection to rotate clockwise when viewed from thenose. As the tail fin approaches the (strongest) NEcanard vortex, the tail rolling moment is strongest(near φtail = 80◦). As the tail section moves tothe + position (φtail = 45◦), the fins are farthestfrom the strong NE canard vortex, and evenly split(vortex suction inducing both CW and C-CW rota-tion) between the remaining two vortices, and thetail rolling moment is nearly the weakest. This dif-ferential pressure on the tail fin in the NE quad-

rant due to the vortices, along with the effects ofdynamic pressure and angle of attack, and the in-duced velocity field from the canards all combine toimpart a net rolling moment on the unconstrainedtail, causing it to spin.

Figure 5: Mach number contours (blue is low, red is high)for a static simulation with tail in × position. The schematicto the right shows the location and sense of rotation of thecanard vortices. (M∞ = 1.6, α = 4.0◦).

0 22.5 45 67.5 90Angle of Rotation from X-configuration

-0.75

-0.5

-0.25

0

Tai

l Rol

ling

Mom

ent C

oeff

icie

nt

Figure 6: Tail rolling-moment for the static, fixed-tail sim-ulations (M∞ = 1.6, α = 4.0◦).

3.2 Forced-spin-rate Tail Section

The rotation rate of the tail section is not knowna priori. In order to determine the “natural” rollrate of the tail section – the rate at which the spin-averaged rolling moment on the tail is zero – an it-erative process is used. First, it’s assumed that thetail rotation rate is low enough that the variationof spin-averaged tail rolling moment with rotationrate is linear. A fixed rotation rate is then imposedon the tail, which is intended to be a reasonableguess. The resulting spin-averaged tail rolling mo-ment from this simulation, along with the staticresults discussed above (i.e. a zero-spin-rate simu-lation) are then fit with a straight line to determine

5

the predicted natural roll rate of the tail section. Asecond dynamic simulation is then performed at thenatural roll rate in order to confirm the prediction.

An initial guess of 2500 rpm for the tail rotationrate was used at α = 4◦. A time-resolution studywas performed at this rotation rate using timestepsthat move the tail fins 2◦, 1◦, 0.5◦, 0.3◦, and 0.15◦

of rotation per step respectively. The results of thistime-resolution comparison showed no difference intail load vs. rotation angle between the 0.3◦ and0.15◦ of rotation/timestep simulations, and as a re-sult 0.3◦ of roll per timestep was utilized for all ofthe simulations discussed here. This timestep issmaller than preferred for computational efficiencyin a dual-time scheme, but is required to accuratelycapture the complex physics with the current nu-merical scheme.

Figure 7 shows the spin-averaged tail rolling mo-ment against the imposed rotation rate for the it-erative process discussed above. The variation ofaveraged tail rolling moment with rotation rate isconfirmed to be linear, and the predicted naturalrotation rate is 2155 rpm for these conditions. Thevariation of tail rolling moment with rotation an-gle is shown in Fig. 8 for all computations; thestatic and the two dynamic with an imposed ro-tation rate. The simulation with the natural ro-tation rate does provide nearly zero spin-averagedtail rolling moment (Cltail = −0.008). As the ca-nard vortices, canard downwash, and wind vectordo not change when the tail spins, the variationof tail rolling moment with rotation angle is simi-lar for all simulations, however shifted as the rota-tion rate increases. This implies that the rotationof the tail section provides minor dynamic effects,and these effects wash downstream without influ-encing the aerodynamic loads. This is especiallytrue with a fixed spin rate as there is no accel-eration of the tail section. When the velocity ofthe tail section “balances” the outer flow effects,a stable spin rate is found. Since the variation intail rolling moment is self-similar with fixed spinrates, it is unnecessary to compute the entire cycleat the initial guess. Once the increment betweenthe static and initial guess is known, i.e. after thetransient portion of the cycle has been computed,this increment can simply be applied to the staticspin-average to obtain the spin-average at the ini-tial guess. This technique was applied successfullyat α = 12.0◦, the results of which will be discussedSec. 3.4.

Mach number contours through the tail section,

0 1000 2000 3000Tail Roll Rate (RPM)

-0.5

-0.4

-0.3

-0.2

-0.1

0

Rol

l-A

vera

ged

Tai

l Rol

ling

Mom

ent C

oeff

icie

nt

Predicted - 2155 RPM

Computed Value @ 2155 RPM = -0.0078

Figure 7: Predicted “natural” tail rotation rate (M∞ =1.6, α = 4.0◦).

0 22.5 45 67.5 90Angle of Rotation from X-configuration

-0.75

-0.5

-0.25

0

0.25

Tai

l Rol

ling

Mom

ent C

oeff

icie

nt

StaticDynamic - 2155 rpmDynamic - 2500 rpm

Figure 8: Tail rolling-moment for static, fixed-tail and dy-namic, forced-spin-rate simulations (M∞ = 1.6, α = 4.0◦).

viewed from the nose, are shown in Fig. 9 for thenatural rotation rate - 2155 rpm. As the fins en-counter the vortices there is a strong interaction,however after the fins pass the vortices reform intheir original positions. A similar series of pres-sure contours on the surface of the tail section isshown in Fig. 10. The shocks emanating from thecomplex geometry upstream interact with the tailsection as it rotates. The narrow gaps and clear-ances in the geometry provide no difficulty for thecurrent method.

3.3 Free-to-spin Tail Section

The forced-spin tail is an approximation to theactual rotation of the tail section. The actual mo-tion will respond to the aerodynamic forces as itspins, and increase or decrease the spin-rate accord-ingly. In order to assess the efficiency and accuracyof the forced-spin approximation, simulations witha free-to-spin tail were performed using the coupledCFD/6-DOF method outlined in [22]. The 6-DOFmotion is constrained to only allow rotation aboutthe longitudinal axis of the missile body, effectively

6

(a) φtail = 0.0◦/90.0◦

(b) φtail = 22.5◦

(c) φtail = 45.0◦

(d) φtail = 67.5◦

Figure 9: Velocity magnitude contours through the tailsection (red is high, blue is low) viewed from the nose(φ̇tail = 2155 rpm, M∞ = 1.6, α = 4.0◦).

(a) φtail = 0.0◦/90.0◦

(b) φtail = 22.5◦

(c) φtail = 45.0◦

(d) φtail = 67.5◦

Figure 10: Surface pressure contours on the tail section(red is high, blue is low). (φ̇tail = 2155 rpm, M∞ = 1.6,α = 4.0◦).

7

limiting this to a 1-DOF simulation. The actual in-ertia of the tail section is unknown, and is approxi-mated by assuming the density of the tail section tobe half the density of aluminum (i.e. assume thatthe tail section is partially hollow). This assumeduniform density is used with the surface geometryto provide an estimate of the rotational inertia re-quired for the 1-DOF simulations (cf. Mirtich[23]).

The 1-DOF simulations were initiated from the2155 rpm forced-spin simulation. As expected, themotion of the tail immediately responds to theaerodynamic forces, and the spin rate of the tail be-gins to vary (cf. Fig. 11). The average spin-rate ofthe tail increases every quarter-revolution the tailcompletes. Note that the 2155 rpm simulation doespredict a small negative tail rolling moment, i.e.predicts that the tail is spinning slightly too slow.The change in average spin-rate each quarter cycleis less than 0.5% of the mean spin rate, and likewisethe maximum deviation from the mean rate overeach cycle is only ±0.5%. Even with these smallvariations in spin-rate, the aerodynamic loads onboth the tail section and the entire missile are pe-riodic after an initial transient lasting roughly 45◦

of tail rotation. The variation of tail rolling mo-ment for the 1-DOF free-to-spin simulation is com-pared to the 2155 rpm forced-spin-rate simulationand the static simulation in Fig. 12. The tail loadvariation closely follows the forced-spin variation,since the tail acceleration is moderate throughout,however the amplitude of the load variation in thefree-to-spin simulation is lower as the tail respondsto the aerodynamic forces.

0 22.5 45 67.5 90Angle of Rotation from X-configuration

2120

2130

2140

2150

2160

2170

2180

Tai

l Ang

ular

Vel

ocity

(R

PM)

First 1/4-RevSecond 1/4-RevThird 1/4-Rev

Figure 11: Tail angular velocity for free-to-spin (1-DOF)simulation. (M∞ = 1.6, α = 4.0◦).

Since the loads on the tail and the entire mis-sile are periodic, even though the spin-rate is stillchanging slightly, meaningful spin-averages can becomputed for comparison with the forced-spin sim-

0 22.5 45 67.5 90Angle of Rotation from X-configuration

-0.25

-0.125

0

0.125

0.25

Tai

l Rol

ling

Mom

ent C

oeff

icie

nt

Fixed Spin - 2155 rpm1-DOF, 1st 1/4-Rev1-DOF, 2nd 1/4-Rev1-DOF, 3rd 1/4-Rev

Figure 12: Tail rolling-moment for forced-spin and free-to-spin simulations (M∞ = 1.6, α = 4.0◦).

ulations. The spin-averaged forces and moments onthe missile for the forced-spin and 1-DOF simula-tions are presented in Table 1. The maximum dif-ference in spin-averaged loads between the forced-spin and 1-DOF simulations is 6%. At these condi-tions the changes in spin-rate over each cycle, andby inference the dynamic effects due to the accel-eration of tail, are both small. Hence the morecomputationally-efficient forced-spin technique canbe used in place of a full free-to-spin simulation.The spin-average can be computed in little morethan a single quarter-revolution with the forced-spin approximation, while the free-to-spin simula-tions may take many revolutions to reach a stablestate.

In order to provide more insight into the aero-dynamics of the spinning-tail configuration, theα = 4.0◦ static and dynamic (both fixed- and free-to-spin) results are analyzed using a plot of thecrossflow moments (Fig. 13). Here the yawing andpitching moments are plotted on the two axes asthe tail spins, along with the spin-averaged val-ues. In order to make performance predictions, thespin-averaged forces and moments of the dynamicconfiguration are required. While one static con-figuration does closely predict the dynamic spin-averaged loads, this is fortuitous, and different ca-nard settings or flow conditions will behave differ-ently. The dynamic curves are shifted up and to theright from the static curve, and maintain the shapebasic shape. This self-similarity is due to the lackof dynamic effects. The tail provides a “restoring”moment opposite to the effect the canards provideahead of the center of mass. When the tail is freeto spin, this restoring moment is reduced as thetail moves in response to the aerodynamic forces.From this point of view, the free-spinning tail be-haves as if it was an equivalent static tail of smaller

8

size. Hence the increment between the static anddynamic simulations is in the direction the canardsare forcing the body to rotate, in this case nose upand starboard.

6 6.2 6.4 6.6 6.8 7<-- Port Yaw Starboard -->

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

<--

nos

e do

wn

Pitc

h

n

ose

up -

-> StaticForced-spin (2155 rpm)Free-to-spin

X

+

Figure 13: Crossflow moments for the static and dynamicsimulations. Spin-averaged values are shown with solid cir-cles. The × and + positions are labeled, along with thedirection of tail spin. (M∞ = 1.6, α = 4.0◦).

Table 1 compares the current inviscid simulationsat α = 4.0◦ with high-resolution, viscous, free-to-spin simulations computed by Nygaard[11] for thesame missile configuration. The computed spin-averaged forces and moments for both free-to-spinsimulations are all within 2.5%, with the exceptionof pitching moment.∗

3.4 Variation with Angle of Attack

The previous sections outline the aerodynam-ics of the free-spinning tail section, along with amethodology for predicting the behavior using aforced-spin approximation at α = 4.0◦. This sec-tion will extend the analysis by considering the be-havior at α = 0.0◦ and 12.0◦. First, static solutionswere computed with the tail in fixed azimuthal lo-cations to complement the results presented previ-ously for α = 4.0◦ in Fig. 6. The variation of tailrolling moment from these series of static simula-tions is presented in Fig. 14. As expected, the com-putations at α = 4.0◦ show the highest magnitudetail rolling moment, as the canard/tail interactionsare strongest at this angle of attack. Correspond-ingly, the variation at α = 4.0◦ never changes sign,so that the tail will spin clockwise regardless of the

∗The differences in axial force are not considered as theinviscid results are not corrected for viscous drag.

initial tail orientation. This differs from the resultsat α = 0.0◦ and 12.0◦, which both show statically-stable tail orientations near φtail = 45.0◦ and 22.5◦

respectively. The spin-averaged tail rolling momentfor α = 0.0◦ is zero since the flowfield is symmet-ric with respect to the horizontal plane, while thespin-averaged tail rolling moment at α = 12.0◦ isnot. Whether these statically-stable positions arealso dynamically-stable to small perturbations isunknown, however it is almost certain that someconditions (angle of attack, canard setting, ...) doprovide regions of local dynamic stability. The im-plications of this will be discussed after the dy-namic simulations at α = 12.0◦ are presented.

0 22.5 45 67.5 90Angle of Rotation from X-configuration

-0.75

-0.5

-0.25

0

0.25

Tai

l Rol

ling

Mom

ent C

oeff

icie

nt

α = 0.0ο

α = 4.0ο

α = 12.0ο

Figure 14: Tail rolling-moment for static simulations at allangles of attack. (M∞ = 1.6).

Since the static results at α = 12.0◦ indicatea non-zero spin-averaged tail rolling moment, theforced- and free-to-spin analysis described in theprevious sections is applied at α = 12.0◦ to pro-vide insight. The forced-spin iterative method pre-dicts a tail spin rate of near 500 rpm at α = 12.0◦.Dynamic forced-spin results were obtained at thisrate, and a 1-DOF free-to-spin simulation was ini-tiated from the fixed-rate computation and contin-ued through two quarter-revolutions. The tail spinrate for the free-to-spin simulation at α = 12.0◦

is shown in Fig. 15, after the initial release fromthe 500 rpm forced-spin simulation. The mean tailspin rate increases by a small amount each quarter-revolution, but in contrast to α = 4.0◦, the rota-tion rate varies by ±10% over each cycle - a factorof 20 more than the variation in Fig. 11. This largevariation in tail spin rate is caused by the torquebeing applied over a longer time duration at thislower spin rate, and indicates that a forced-spinsimulation at a fixed rate is not a good approxima-tion for the actual motion at this angle of attackif the tail is spinning. Note that even though the

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CA CY CN Cl Cm Cn

Static 0.48 0.65 1.11 -0.40 -0.22 6.45Fixed-spin (2155 rpm) 0.53 0.61 1.12 -0.01 -0.16 6.55

Free-to-spin 0.50 0.61 1.12 0.00 -0.17 6.57Free-to-spin Ref. [11] 0.69 0.60 1.15 -0.01 -0.04 6.49

Table 1: Computed spin-averaged forces and moments on the entire missile. All values are provided using standardaircraft coordinates. Axial force does not include the missile base section. (M∞ = 1.6◦, α = 4.0◦).

static results indicate a statically-stable tail orien-tation, the mean tail rotation rate still increases.The large variation in tail spin rate implies thatthe simulation contains large dynamic effects. Thisis seen in the crossflow moments plotted in Fig. 16.The static and dynamic forced-spin results are veryclose, as anticipated with this low spin rate, how-ever the free-to-spin simulation predicts a large in-crement, and no longer maintains a shape similarto the static result. This change in shape is causedby the accelerations on the tail altering the dy-namic flowfield from the static or forced-spin re-sults. Since the shape of the crossflow plot canchange almost arbitrarily under these conditions itis not reliable to use a simpler approximation, suchas static or forced-spin, and apply a post-processingcorrection.

0 22.5 45 67.5 90Angle of Rotation from X-configuration

400

425

450

475

500

525

550

Tai

l Ang

ular

Vel

ocity

(R

PM)

First 1/4-RevSecond 1/4-Rev

Figure 15: Tail angular velocity for free-to-spin (1-DOF)simulation. (M∞ = 1.6, α = 12.0◦).

The behavior of the tail rolling moment withvarying angle of attack indicates that the actualbehavior of the tail section is likely multi-valued.The results at α = 12.0◦ demonstrate the possibil-ity that a fixed tail orientation can be stable, whileat the same time indicate strongly that an initiallyspinning configuration may not decay (stop spin-ning) at this same angle of attack. This impliesthat the state of the tail section at any time, andby inference the loads on the missile, is dependentupon the past time history of the missile configu-

4 4.5 5 5.5 6 6.5<-- Port Yaw Starboard -->

-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

<--

nos

e do

wn

Pitc

h

n

ose

up -

-> StaticForced-spin (500 rpm)Free-to-spin

X

+

Figure 16: Crossflow moments for the static and dynamicsimulations. Spin-averaged values are shown with solid cir-cles. The × and + positions are labeled, along with thedirection of tail spin. (M∞ = 1.6, α = 12.0◦).

ration. Depending upon the initial state and thepath through time, different behavior can be ob-served. For example, in the current configuration atα = 12.0◦, if the canards are initially undeflected,the flow will be symmetric and the tail will notbe spinning. If the canards are then deflected, it’splausible that the tail section will remain fixed atthe static stability point. However, if the canardsare initially deflected at α = 4.0◦ so that the tail isspinning at a high rate, and the body is pitched upto α = 12.0◦, then it is plausible that the tail willremain spinning. Without knowledge of this timehistory it cannot be predicted in general whetherto use a static analysis, a forced-spin analysis, ora full free-to-spin simulation. Further, the state ofthe spinning tail is also dependent upon its iner-tial properties. Note that this requires the inertialproperties of a wind tunnel model to be scaled ap-propriately to predict the correct dynamic behaviorof the actual system. Based upon the current re-sults, the tail section can remain fixed, can rotateconstantly, or could continually oscillate betweenthe extremes of fixed and rotating tails. All of thesebehaviors have been observed during unpublishedexperimental tests of the current configuration.

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4 Summary

The canard-controlled missile with a free-spinning tail section considered in the current workprovides a difficult challenge for CFD methods.The geometry is extremely complex, and containsmany scales and small clearances which must beresolved. The physics of the canard/tail interac-tion requires a significant volume mesh resolution,which places an emphasis on the efficiency of anysolution process. Finally, in general the dynamictail section requires a moving-body flow solver cou-pled with a 6-DOF module in order to providean accurate analysis. An automated 3-D Carte-sian method for analyzing complex geometries withcomponents in relative motion has been demon-strated. The complete CFD solution process is au-tomated: a surface triangulation is automaticallygenerated from a CAD solid model, the volumemesh is automatically generated using the Carte-sian scheme, and the dynamic flow solver is robustand efficient, and can handle any general rigid-bodymotion through a simple interface.

Three levels of fidelity were applied to analyzethe missile with spinning-tail section at M∞ =1.6◦: a series of static simulations with the tailin fixed azimuthal orientations, a forced-spin dy-namic simulation using an iterative approach to de-termine an appropriate fixed spin-rate, and a free-to-spin simulation using a flow solver coupled witha constrained 6-DOF module. The analysis of thespinning-tail configuration indicates that the choiceof a static, forced-spin, or free-to-spin analysis can-not in general be made a priori. Attempting to cor-rect low-fidelity results in a post-processing phaseis also not possible in general due to the large dy-namic effects which the spinning-tail configurationcan generate. A variation in tail rolling momentcan indicate regions of static stability, however thissame variation can imply that an initially spinningtail will continue to spin. Further, the behaviorof the dynamic tail section is likely multi-valued,and hence the state for any configuration is depen-dent upon the past history of the missile and itsactual inertial properties. The investigation indi-cates that the tail section can remain fixed, canrotate constantly, or could continually oscillate be-tween the extremes of fixed and rotating tails, andthat all of these behavior could plausibly occur atthe same flow conditions and control surface set-tings depending upon the previous history. Thedetailed information from a CFD analysis provides

the insight which is required to understand, andultimately predict, the behavior of such a complexsystem.

Acknowledgments

The authors would like to thank Dr. Tor Nygaardof ELORET for providing the results of the vis-cous simulations, and Dr. Robert Meakin of theU.S. Army AFDD for providing the software tocompute the inertial properties of the tail section.The authors would also like to thank Prof. Mar-sha Berger of the Courant Institute for her manyinsightful discussions on Cartesian methods.

References

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[18] Jameson, A., “Time Dependent CalculationsUsing Multigrid with Applications to Un-steady Flows Past Airfoils,” AIAA Paper 91-1596, June 1991.

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[22] Murman, S.M., Aftosmis, M.J., and Berger,M.J., “Simulations of 6-DOF Motion with aCartesian Method,” AIAA Paper 2003-1246,Jan. 2003.

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