An Aerodynamic Analysis of a Spinning Missile with Dithering Canards

29 Pages·2003·1.6 MB·English

by  Meakin, Robert L.

An Aerodynamic Analysis of a Spinning Missile with Dithering Canards
Tor A. Nygaard* and Robert L. Meakin†
Army/NASA Rotorcraft Division
NASA Ames Research Center
Moffett Field, CA 94035-1000
Abstract cousandmissileroll-rateeffects. Theaerodynamicper-
Agenericspinningmissilewithditheringcanardsis formanceofthemissileasafunctionofangle-of-attack
used to demonstrate the utility of an overset structured and canard pitching sequence is also considered. A
grid approach for simulating the aerodynamics of novel analytical method for describing these perfor-
rolling airframe missile systems. The approach is used mance characteristics is given. A brief summary,
to generate a modest aerodynamic database for the acknowledgements, and list of references are provided
generic missile. The database is populated with at the end of the paper.
solutionstotheEulerandNavier-Stokesequations. Itis
used to evaluate grid resolution requirements for
accurate prediction of instantaneous missile loads and 2. ROLLING AIRFRAME CONFIGURATION
therelativeaerodynamicsignificanceofangle-of-attack,
canard pitching sequence, viscous effects, and roll-rate The generic rolling airframe missile employed in
effects. A novel analytical method for inter- and thepresentworkisreferredtoasFM-3. TheFM-3mis-
extrapolation of database results is also given. silehasahemisphericalnose,cylindricalbody,fourfins,
andtwocanards. Detailsofthegeometriccomplexityof
the missile are illustrated in Figure 1.
1. INTRODUCTION
Simulation of the aerodynamics of rolling airframe
missilesystemsposesignificantchallengesforanycom-
a)
putational approach. Applications of practical interest
are characterized by complex vortical flow and shock
structures. In addition, the geometry of these missile groove
systemscanbeverycomplex,involvingrelativemotion
between missile body and control surfaces. The flows
arealsoinherentlyunsteady. Theaimofthispaperisto b)
demonstrate the utility of Chimera1 overset structured
shaft
grid domain decomposition methods in the efficient
generation of high fidelity aerodynamic simulations for
this class of problems.
cut-out
A generic rolling airframe missile is defined in
order to demonstrate the advantages and limitations of
c)
an overset grid approach. The paper provides a techni-
cal description of the generic missile and the specific
computational methods here employed, and complete
discussions of the case conditions and corresponding
simulation results. The set of simulations considered
are designed to demonstrate the level of resolution
Figure 1. FM-3 missile geometry. a) Top view of
required for accurate prediction of surface loads and to
entire missile. b) Fins. c) Close-up of canard.
determine the relative aerodynamic significance of vis-
* Research Scientist, ELORET Corp., Member AIAA
† Senior Research Scientist, U.S. Army AFDD (AMCOM), Senior Member AIAA
ThismaterialisdeclaredaworkoftheU.S.Governmentandis
not subject to copyright protection in the United States.
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Thefinsaredesignedtoinducemissilespin,while tance away from the respective surfaces. The
directional control is actuated via canard dithering. As constructionofnear-bodygridsandassociatedintergrid
the missile spins, the canard pitch position follows an connectivity is a classical Chimera-style decomposition
actuator signal with constant pitch rate. The actuator of the near-body domain. It is assumed that near-body
signal flip-flops between +/- 1 according to the sign of grids provide grid point distributions of sufficient den-
the sum of two sine-waves called the command and sity to accurately resolve the flow physics of interest
dither signals. The amplitude of the command signal (i.e., boundary-layers, vortices, etc.) without the need
relativetothedithersignaliscalledthecommandlevel, for refinement. This is a reasonable constraint since
andreflectsthestrengthoftheattemptedmaneuver. The near-body grids are only required to extend a short dis-
command signal is modulated with the roll-rate. The tanceawayfrombodysurfaces. Figure3illustratesthe
dithersignalismodulatedwithaditherfrequency. Fig- surface decomposition of the FM-3 missile and shows
ure 2 shows the canard pitching algorithm for a com- selected surfaces from the resulting near-body surface
mandlevelof100%forthespecifiedroll-rateanddither and volume grids.
frequency.
The off-body portion of the domain is defined to
encompass the near-body domain and extend out to the
3. COMPUTATIONAL METHODS far-field boundaries of the problem. The off-body
domain is filled with overlapping uniform Cartesian
3.1 Discretization Method grids of variable levels of refinement, as shown in Fig-
ure4fortheFM-3missile. Theoff-bodygridresolution
The “near-body” and “off-body” domain partition- amplificationfactorbetweensuccessivelevelsis2. The
ingmethoddescribedinReferences2and3isusedhere near-bodyoff-bodypartitioningapproachfacilitatesgrid
asthebasisofdiscretizationoftheFM-3missile. Inthe adaptationinresponsetoproximityofbodycomponents
approach, the near-body portion of a domain is defined and/ortoestimatesofsolutionerrorwithinthetopologi-
toincludethesurfacegeometryofallbodiesbeingcon- cally simple off-body grid system.
sidered and the volume of space extending a short dis-
Canard Pitching Algorithm
2
Command
Dither
1.5
Command+Dither
−]
h [ Canard Pitch
c 1
Pit
d
e
z
ali 0.5
m
r
o
N
d 0
n
a
s
al−0.5
n
g
Si
ol
r −1
nt
o
C
−1.5
−2
0 30 60 90 120 150 180 210 240 270 300 330 360
Roll Angle [Degrees]
Figure 2. Canard pitching algorithm for roll-rate of 8.75 Hz, dither frequency of 35 Hz, and 100% command level.
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Level-3
a)
Level-2
Level-1
b)
near-body grid
components
Figure 4. FM-3 near-body and off-body partitioning
and selected surfaces.
3.2 Solution Method
The set of FM-3 missile simulations presented in
thispaperrepresentawidevarietyofconditionsandare
productsoftheOVERFLOW-D3,5code. OVERFLOW-
D is based on version 1.6au of the well known NASA
OVERFLOW6 code, but has been significantly
enhanced to accommodate moving body applications.
The OVERFLOW-D enhancements represent in-core
subroutine actuated operations and include the follow-
ing capabilities.
i. On-the-flygenerationofoff-bodygridsystems.
ii. MPI6 enabled scalable parallel computing.
c)
iii. Automatic load balancing.
iv. Aerodynamic force and moment computations.
v. General 6-degrees-of-freedom model.
vi. Rigid-body relative motion between an arbi-
trary number of bodies.
vii. Domain connectivity.
viii.Solution error estimation.
ix. Grid adaptation in response to body motion
and/or estimates of solution error.
ThemajorityoftheFM-3simulationspresentedin
this paperinvolve relativemotion betweengrid compo-
Figure 3. FM-3 surface geometry decomposition and
nents. The entire missile spins relative to the inertial
near-body grids. a) Surface decomposition of missile
off-body grid system and the canards dither relative to
body. b) Decomposition of canard surfaces and
the missile body. The pseudo-code below outlines the
selectedsurfacesfromcorrespondinggrids. c)Decom-
general procedure used in OVERFLOW-D to carry out
positionoffinsurfacesandselectedsurfacesfromcor-
such simulations. Of course, the flow equations are
responding grids.
solved at every time-step during a simulation. In cases
Note: Surfacegeometrydecompositionandnear-bodygridgenera- thatinvolverelativemotionbetweenconfigurationcom-
tion accomplished using OVERGRID utility from CGT4. ponents, body dynamics and domain connectivity are
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alsoaddressedateachtime-step. InthecaseoftheFM- when each processor assumes the load of at least 250
3 missile, body dynamics simply means the computa- thousand points. Load balancing is an automatic func-
tionofaerodynamicloadsandmovingthemissilecom- tion of OVERFLOW-D.
ponents according to a control-law. Specifically, the
rotational orientation of the missile is positioned as a As indicated in the pseudo-code above, OVER-
function of time and the roll-rate. The position of the FLOW-D accommodates solution adaptation based on
canardsrelativetothemissilebodyisdeterminedbythe the position of near-body grid components and/or in
canard dither algorithm (illustrated in Figure 2). response to estimates of solution error. The off-body
gridmanagementschemeallocateslevel-1(finest)reso-
lutiongridstoaccommodatesignificantmotionofbody
For N time-steps
componentsorflowfeaturesbeforethenextadaptcycle.
do every step Accordingly, adapt cycles are only required periodi-
cally; every 25 to 50 time-steps in a typical unsteady
Solve flow equations
simulation.
For Moving Body Problems
- Body dynamics
IntheFM-3missilecasesconsideredhere,allflow
- Domain connectivity features that are likely to have any significant affect on
thesurfaceforcesandmomentsareconfinedtothevol-
do every mth step ume of space within a missile diameter of the body
itself. These include canard vortices, boundary layer,
Adaptive Refinement
and key portions of the shock systems. Accordingly,
- Error estimation OVERFLOW-Dinputisusedtoallocatelevel-1resolu-
- Off-body re-partitioning tion capacity to a distance of 1.5 diameters from the
- Solution transfer missile surface, rather than enable adaptation in
- Domain connectivity responsetosolutionerror. Aslightsavingsincomputa-
tional overhead is thereby gained for the present cases.
Pseudo-Code. Solution procedure (with adaptive
refinement capability) for unsteady problems that may
involve relative motion between component parts. 4. SIMULATION RESULTS
A set of FM-3 missile simulations is carried out to
Since the missile movement is continuous, the demonstratethelevelofresolutionrequiredforaccurate
relativepositionofmanygridcomponentschangeevery prediction of surface forces and moments and to deter-
time-step. In order for solution information to be cor- mine the relative aerodynamic significance of viscous
rectly exchanged between grids during the simulation, effects, missile roll-rates (W ), canard command levels
r
the domain connectivity solution must also be continu- (c), and free-stream angles-of-attack (a ). A total of 31
ously updated. This is accomplished automatically by FM-3 simulations form the basis of the material pre-
OVERFLOW-D. sentedinthispaperconcerningtheseissues. Theflight
conditions for the cases are indicated in Table 1. The
The OVERFLOW-D processing rate for static parameters varied to obtain the complete simulation set
geometryviscous flowapplications isabout 15m secper are free-stream angle-of-attack, canard pitch command
grid-pointpertime-step(300MHzprocessor). Formov- level,andmissileroll-rate. Theothersimulationparam-
ing-body problems, the processing rate is somewhat eters indicated in the table are held fixed and are com-
problemdependent,butgenerallyfallsintheboundsof mon to all cases considered.
15to18m secpergrid-pointpertime-step. FortheFM-
3spinningmissilecasesconsideredhere,thenumberis
Table 1. Simulation Parameters
16.5m secpergrid-pointpertime-step. OVERFLOW-D
accommodates problem sizes of more than 2 million M¥ Mach number 1.6
grid-pointsper1gigabyteofmemory. Maximumparal- Re Reynolds number 50· 106
lelefficiency(percentageinhigh90’s)isrealizedwhen W Dither-frequency 35 Hz
d
the fewest number of processors that can accommodate a Angle-of-attack 0o, 2o, 3o, 4o, 8o, 12o, 15o
a given problem in core memory are selected. OVER-
c Command level 0%, 100%, 200%
FLOW-D can efficiently (i.e., over 70%) make use of
W Roll-rate 0 Hz, 8.75 Hz
larger numbers of processors for a fixed problem size r
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The general characteristics of the FM-3 flow field 3o, and 8.75 Hz, respectively. A very high resolution
are illustrated in Figure 5. Vortices are shed from the gridisusedtodefinethebaselinesolution. Mediumand
inboard and outboard tips of the canard and convect coarse solutions are obtained on grids derived from the
down the length of the missile interacting with the vis- baseline grid with successively lower levels of spatial
cous boundary layer. Away from the influence of the resolution. The qualitative effect of coarsening on the
boundary layer, the outboard canard vortices twist surface geometry is shown in Figure 6. A very high
approximately 8o around the spinning missile in one fidelitytemporal resolution(viz.,12,000time-stepsper
body-lengthoftravel. Ascanbeseeninthefigure,dis- missile revolution) is uniformly employed in all of the
ruptions to the boundary layer by the inboard canard viscous simulations.
vorticesare draggedthroughnearly45oofrolloverthe
sameinterval. TheshockstructureisindicatedinFigure
5b. Theboundarylayergrowthontheuppersurfaceof
themissileisalsovisibleinthefigure. Apositiveangle-
of-attackandvortex/boundarylayerinteractioncombine
to exaggerate the boundary layer thickness down the
stream-wise axis of the missile.
V1 (fine)
Outboard canard
Inboard canard vortex
vortex
V2 (medium)
a)
V3 (coarse)
Vortex/Boundary layer
interaction Figure 6. Fine, medium, and coarse grid representa-
tions of FM-3 canards and fins.
The baseline grid for this case (finest resolution) is
comprised of 41 million grid points and is referred to
hereafterastheV1(i.e.,“Viscous-1”)grid. Isolatedsur-
faces from the V1 grid are shown in Figures 3 and 4.
b)
The flowcharacteristics illustratedin Figure5 arefrom
Figure5. AerodynamicsoftheFM-3spinningmissile asimulationusingtheV1grid. Allgridlengthsreferred
with dithering canards. a) Vortex structure. b) Shock to in the following discussion are normalized by the
structure. W r = 8.75 Hz,c = 0%,a = 3o. missile body length. Viscous spacing normal to the
bodysurfacesis 2.5· 10–6 intheV1grid. Thiscorre-
spondstoay+of1foraReynoldsnumberof10million.
4.1 Resolution Requirements
This spacing is maintained uniformly across the first 6
cells in the viscous direction and then expanded with a
A grid refinement study is used to determine the
geometricstretchingratioof1.2toadistanceofapprox-
level of spatial resolution needed to accurately predict
imately0.015. Themaximumspacingusedinthenear-
the integrated FM-3 surface loads. The significance of
body grids is approximately equal to the level-1 off-
grid resolution is evaluated here by comparing viscous
body grid spacing which is 0.0013, or approximately
solutionsforthespinningmissilecasedefinedinTable1
0.1% of the body length.
with the variable parameters c, a , and W fixed at 0%,
r
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The V1 grid is the basis of the medium (V2) and Table 2. Roll-averaged force and moment coefs.*
coarse (V3) grids. The V2 grid is obtained by deleting
Coefficient Fine (V1) Med. (V2) Coarse (V3)
approximatelyeveryotherpointfromtheV1gridinall
Cx (axial) 1.17 1.12 1.07
three spatial dimensions and results in a grid with just
Cy (side) -7.56e-03 2.76e-03 -1.21e-03
over8millionpoints. Similarly,theV3gridisobtained
Cz (normal) 0.461 0.462 0.538
fromtheV2gridbydeletingapproximatelyeveryother
Cmx (roll) -1.19e-03 -1.08e-03 -0.94e-03
point from the V2 grid in all three spatial dimensions
Cmy (pitch) -6.79e-03 -3.11e-03 -3.45e-02
andresultsinagridwithjustover2millionpoints. The
Cmz (yaw) 2.80e-03 3.52e-03 -9.55e-04
foregoing is true subject to the following qualifications.
i. Some surface grids require redistribution and/ *Moments are about the missile center of gravity
oradditionofgrid-pointstopreservegeometric
Figure9),indicateasystematicshifthigherforfinerres-
features such as sharp corners through the two
olution. The difference between V1 and V2 axial
subsequent eliminations of every other point.
forces is approximately the same as between the corre-
ii. Smoothingisappliedtogeometricfeaturesthat
spondingV2andV3results. Clearly,gridconvergence
are not adequately resolved by the coarser
is not apparent in the computed axial force data.
grids. FortheV2grid,smoothingisappliedto
thecanardcut-outandthemissilegroove. For
Figures10and11breakdowntheaxialforcesinto
the V3 grid, the canard cut-out, the canard
pressureandviscouscomponents. TheV2andV3solu-
shaftandthemissilegrooveareremovedcom-
tions are almost identical for the pressure component;
pletely.
and V1 has a systematic shift to a higher value. The
iii. The grid spacing in the surface normal direc-
contributiontothisshiftcomesmainlyfromtheaftpart
tionfortheV2gridcorrespondsapproximately
ofthe missile. Thisregionofthe flowhascomplicated
to every other point for the V1 grid, doubling
interactions between the expansion waves around the
the initial spacing from the wall.
boat-tail, shocks around the tail fins, and the boundary
iv. The grid spacing in the surface normal direc-
layer. The V1 and V2 viscous components to the axial
tion for the V3 grid starts at the surface with
forcedifferbyapproximately2%ofthetotalaxialforce,
the V2 spacing doubled. The stretching ratio
slightlylessthanthedifferencebetweentheV2andV3
thereafter is approximately the same as in the
results. The total roll-averaged axial forces are shown
V2 grid.
inTable2. Themediumandcoarsesolutionaxialforces
are 4% and 8% lower than for the corresponding fine
Considernowthecomputedloadhistoriesobtained
solution. The reason grid convergence in axial force is
from simulations using the V1, V2, and V3 grids. Fig-
not demonstrable via the current set of solutions is not
ure7showsthecomputednormalforce(Cz)historyfor
clear. ItmaybethatwhiletheV1gridhassufficientres-
the three different resolution capacities. The canard
olution in the boundary layer, the V2 and V3 viscous
pitch angle history is also indicated. The V1 and V2
spacing (double and quadruple that of V1) is not suffi-
results are in good agreement, except at maximum
cient.
canard deflection. At high canard lift, the strong vorti-
cesshedfromthecanardsmodifythepressuredistribu-
Figure12showsthepitchingmoment(Cmy)about
tion on the fuselage and the tail fins. Still, the roll-
thecenterofgravityfortheV1,V2,andV3simulations.
averagednormalforcesfromV1andV2showninTable
Themissilecenterofgravityislocatedapproximatelyat
2 differ by less than 0.3%, indicating near grid conver-
the missile midpoint. As is the case for normal force,
gence for this quantity. The V3 result differs signifi-
the differences in pitching moment are largest at maxi-
cantly from V1 and V2.
mum canard deflection. Still, the overall agreement is
verygood. ThedifferencebetweentheV1andV2roll-
ThedatarepresentedinFigures8and9observethe
averagedpitchingmomentsareapproximately1%ofthe
sameformasthatusedinFigure7. However,Figures8
maximum pitching moment during a revolution. The
and9displaysideforce(Cy)andaxialforce(Cx)histo-
roll-averaged pitching moments are shown in Table 2.
ries, respectively. The side forces exhibit the same
The percentage of maximum pitching moment is used
effect as for the normal forces at maximum canard
here as a measure of grid convergence since the roll-
deflection. The roll-averaged side-forces are close to
averaged moments are all nearly zero.
zero,withadifferencebetweenV1andV2oflessthan
0.02% of the normal force, indicating grid convergence
Figure 13 shows the yawing moment (Cmz) about
for this quantity also. In contrast, the axial forces (see
thecenterofgravity. Theagreementisverygood,with
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Normal Force Coefficient
1.2 15
Fine
Medium
Coarse
1
Canard Pitch
0.8 s]
e
e
gr
e
0.6 D
−] h [
z [ 0 Pitc
C 0.4 ard
n
a
C
0.2
0
−0.2 −15
00 3300 6600 9900 112200 115500 118800 221100 224400 227700 330000 333300 336600
−Roll Angle [Degrees]
Figure 7. Grid effects in theNORMAL FORCE coefficient.
Side Force Coefficient
0.8 15
Fine
Medium
0.6
Coarse
Canard Pitch
0.4
s]
e
e
0.2 gr
e
D
−] h [
y [ 0 0 Pitc
C d
ar
−0.2 an
C
−0.4
−0.6
−0.8 −15
00 3300 6600 9900 112200 115500 118800 221100 224400 227700 330000 333300 336600
−Roll Angle [Degrees]
Figure 8. Grid effects in the SIDE FORCE coefficient.
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Axial Force Coefficient
1.4 15
1.2
1 s]
e
e
gr
e
0.8 D
−] h [
x [ 0 Pitc
C0.6 ard
n
a
C
0.4
Fine
0.2
Medium
Coarse
Canard Pitch
0 −15
00 3300 6600 9900 112200 115500 118800 221100 224400 227700 330000 333300 336600
−Roll Angle [Degrees]
Figure 9. Grid effects in the AXIAL FORCE coefficient.
Pressure Axial Force Coefficient
1.2 15
Fine
Medium
1.15 Coarse
Canard Pitch
1.1
s]
e
e
1.05 gr
e
D
−] h [
x [ 1 0 Pitc
C d
ar
0.95 an
C
0.9
0.85
0.8 −15
00 3300 6600 9900 112200 115500 118800 221100 224400 227700 330000 333300 336600
−Roll Angle [Degrees]
Figure 10. Grid effects in the AXIAL FORCE coefficient (PRESSURE component).
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Viscous Axial Force Coefficient
0.2 15
Fine
Medium
Coarse
Canard Pitch
0.15
s]
e
e
gr
e
D
−] h [
x [ 0.1 0 Pitc
C d
ar
n
a
C
0.05
0 −15
00 3300 6600 9900 112200 115500 118800 221100 224400 227700 330000 333300 336600
−Roll Angle [Degrees]
Figure 11. Grid effects in the AXIAL FORCE coefficient (VISCOUS component).
Pitching Moment Coefficient − Cg
0.4 15
Fine
Medium
Coarse
Canard Pitch
0.2
s]
e
e
gr
e
D
my [−] 0 0 Pitch [
C d
ar
n
a
C
−0.2
−0.4 −15
00 3300 6600 9900 112200 115500 118800 221100 224400 227700 330000 333300 336600
−Roll Angle [Degrees]
Figure 12. Grid effects in the PITCHING MOMENT coefficient.
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Yawing Moment Coefficient − Cg
0.4 15
Fine
Medium
Coarse
Canard Pitch
0.2
s]
e
e
gr
e
D
mz [−] 0 0 Pitch [
C d
ar
n
a
C
−0.2
−0.4 −15
00 3300 6600 9900 112200 115500 118800 221100 224400 227700 330000 333300 336600
−Roll Angle [Degrees]
Figure 13. Grid effects in the YAWING MOMENT coefficient.
some minor differences at maximum canard deflection. at maximum negative canard deflection). The position
The difference between the V1 and V2 roll-averaged ofthevorticesareingoodagreementfortheV1andV2
yawing moments are approximately 0.2% of the maxi- solutions. However, the vortex strength is weaker and
mum pitching moment during a revolution. The roll- interactionsbetweentheinboardcanardvortexandvis-
averaged yawing moments are shown in Table 2. cous boundary layer of the missile body are less appar-
ent in the V2 solution. The V3 solution differs
The rolling moment (Cmx) is shown in Figure 14. significantly from the V1 and V2 solutions in vortex
As expected, the moments are small. The difference position, strength, and vortex/boundary layer interac-
between the V1 and V2 roll-averaged rolling moments tion.
are approximately 0.03% of the maximum pitching
momentduringarevolution. Asteady-statefreelyspin- Thegridrefinementresultssuggestverygoodover-
ningmissileshouldhaveanaveragedrollingmomentof allagreementbetweentheV1andV2solutions,though
zero. Theoverallpatternhereisinfairagreement,but somedifferencesareapparent. Still,theV2gridoffersa
the details differ significantly. The rolling moment is good compromise between solution accuracy and solu-
sensitive to flow details around the tail fins, with com- tion throughput for computations designed to predict
plexinteractionsbetweentheboundarylayer,thecanard aerodynamicforcesandmoments. Asnotedabove,grid
vortices and the shocks, which again depend on the convergence of all forces and moments (except axial
accurate prediction of the flow along the entire missile. force and rolling moment) are obtained.
Grid convergence for so small a quantity as the rolling
moment is beyond the capacity of the present set of 4.2 Viscous Effects
grids.
A comparative evaluation of very high resolution
Theeffectofgridresolutiononthepredictionofthe Navier-Stokes and Euler simulations is used to deter-
aerodynamic details of the flow is illustrated in Figure mine the relative significance of viscous effects opera-
15. Figures 15a, 15b, and 15c provide a comparative tive in the range of flight conditions considered for the
view of the canard vortices and missile boundary layer FM-3 missile. The case conditions defined in Table 1
interactionviaplotsofhelicitydensity(i.e.,dotproduct aretakenasrepresentativeoftheseflightconditions. A
of the velocity and vorticity vectors) at several stations static FM-3 case is first considered where the missile
alongthelengthofthemissile(foraroll-angleof268.5o roll-rateiszeroandthecanardsarefixedinneutralposi-
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