INTERNA nONAL
JOURNAL FOR NUMERICAL METHODS IN FLUIDS, VOL. 24,
1353-1369
(1997)
PARALLEL FINITE ELEMENT SIMULATION OF LARGE
RAM-AIR PARACHUTES
V. KALRO1, S. ALIABADII,
W. GARRARD I , T. TEZDUYAR1*,
S. MITTAL2 AND K. STEIN3
IAerospace Engineering and Mechanics, AmlY HPC Research Center, 1100 Washington Avenue South, University of
Minnesota, Minneapolis, MN 55415, U.S.A.
21ndian Institute of Technology, Kanpur, India
3U.S. AmlY Natick RD&E Center, Natick, U.S.A.
SUMMARY
In the near future, large ram-air parachutesare expectedto provide the capability of delivering 21 ton payloads
from altitudes as high as 25,000 ft. In developmentand test and evaluation of theseparachutesthe size of the
parachuteneededand the deploymentstagesinvolved make high-performancecomputing (HPC) simulationsa
desirablealternative to costly airdrop tests. Although computationalsimulations basedon realistic, 3D, timedependentmodels will continue to be a major computationalchallenge, advancedfinite element simulation
techniquesrecently developedfor this purposeand the execution of these techniqueson "PC platforms are
significantstepsin the direction to meetthis challenge.In this paper,two approachesfor analysisof the inflation
and gliding of ram-air parachutesare presented.In one of the approachesthe point mass flight mechanics
equationsare solved with the time-varying drag and lift areasobtained from empirical data. This approachis
limited to parachuteswith similar configurationsto thosefor which dataare available.The other approachis 3D
finite elementcomputationsbasedon the Navier-Stokesequationsgoverning the airflow around the parachute
canopy and Newton's law of motion governing the 3D dynamicsof the canopy, with the forces acting on the
canopy calculated from the simulated flow field. At the earlier stagesof canopy inflation the parachuteis
modelled as an expandingbox, whereasat the later stages,as it expands,the box transformsto a parafoil and
glides. Thesefinite elementcomputationsare carried out on the massivelyparallel supercomputersCRAY T3D
and Thinking MachinesCM-5, typically with millions of coupled, non-linear finite elemeJItequationssolved
simultaneously at every time step or pseudo-time step of the simulation.
~
1997 by John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Fluids, 24: 1353-1369, 1997
No. of Figures: 20.
No. of Tables: O.
No. of References:21.
KEYWORDS: parallel computations; parachutes; 3D flow simulations
I. INTRODUCTION
Small-scalegliding (ram-air) parachutes,commonly usedby the sportsparachutecommunity as well
as by the military for personneldrop, have reacheda satisfactorylevel of reliability, aerodynamic
efficiency and controllability. Furthermore, larger-size versions of these parachutesare being
increasingly used for the recovery of large payloads.I Gliding parachutescoupled with automatic
onboardguidanceoffer superiorcontrollability and substantialwind penetrationwhen comparedwith
*ColTespondence to: T. Tezduyar, Aerospace Engineering and Mechanics, Army HPC Research Center, 1100 Washington
Avenue South, University of Minnesota, Minneapolis, MN 55415, U.S.A.
CCC 0271-2091/97/121353-17$17.50
@ 1997by John Wiley & Sons,Ltd.
Received17 January 1996
Revised 14 Fl!hrunrv
IQQfI
1354
V. KALRO ET AL
round parachutes.Since their introduction in the early 1960s, gliding parachuteshave been
redesignedand refinedby the sportscommunity. Thesepersonnel-typeparachutesare small but have
lower wing loading than thoserequired for large payloads.
Future military airdrops will require the deploymentof high-altitude delivery systemscapableof
delivering up to 21 tons from 25,000 ft above ground level with increasedaccuracyand reduced
impact velocity. Gliding parachuteswhich are at least an order of magnitudelarger and with a wing
loading three times larger than existing parachutesare currently being developed.
The deploymentand control of such large parachutesposemany challengingtechnical problems.
In the designof any parachutesystemit is important to predict openingforcesfor choiceof materials.
Only a limited databaseis available for large gliding parachutes;therefore methodsfor inflation
analysisbasedon first principles may be useful in design.
In this paper,two approachesfor openingforce analysisare presented.2.3
One of the approachesis
an extensionof the classicalPflanz method.4-6Here the lift and drag areasare assumedto vary with
time and the point mass flight mechanicsequationsare solved as a function of time to yield the
openingforces.This methodrequiresempirical data in order to model the temporal evolution of lift
and drag, so its predictive capabilitiesare restrictedto parachutessimilar to thosefor which data are
available.
The second approach focuses on advanced finite element flow simulation techniques to use
realistic, 3D computer models representingthe parachutesystem and its deployment stages.The
aerodynamics of ram-air parachutes involves a large number of complex phenomena. The
deployment, extension and evolution to the gliding stage involve rapidly changing geometries,
unsteadyand turbulent flow behaviourand non-linear interactionsbetweenthe parachutestructure,
aerodynamicforces and the payload. Even the gliding stageinvolves deformationsof the canopy,
changesin the orientation of the parachuteand changesin relative motion betweenthe canopy and
the payload.All thesebehavioursrequire 3D simulationtechniquescapableof handling time-varying
computationaldomainswith formulations which are robust and accurate.Implementationson HPC
platforms with sufficient computationalspeedand memory are necessary.
At this phaseof the simulationsthe time-variation of the geometryof the canopyis assumedto be
given, approximated using the initial and final configurations and dimensions of the canopy.
However, the dynamicsof the canopy, i.e. its translationaland rotational motion, still needsto be
determinedas part of the overall solution. This motion dependson the weight and motion of the
payloadand also on the aerodynamicforcesgeneratedby the unsteadyflow field. The airflow around
the parachutecanopy is governedby the 3D Navier-Stokes equationsof incompressibleflow with
time-dependentspatial domains.The 3D dynamics of the canopy is governedby Newton's law of
motion, with the forces acting on the canopycalculatedfrom the simulatedflow field. At the initial
stageof canopyinflation the parachuteis modelled as a falling, expandingbox, whereasat the later
stagesthe box transformsto an expanding,gliding parafoil. It is important to note that to accurately
resolvethe flow aroundsuchcomplex 3D geometries,it is essentialto usevery refinedcomputational
grids leading to very large systemsof non-linear equations.The availability of advancedHPC
platforms and efficient implementationtechniquesmakesthesecomputationsfeasible!'s
To handlethe time-variant domainsencounteredin simulationsof parachutesystems,we employ
the deformable-spatial-domainjstabilized-space-time
(DSDjSST) finite elementformulation. In this
formulation the finite elementinterpolationpolynomials are functionsof both spaceand time and the
stabilized variational formulation of the problem is written over the associatedspace-timedomain.
This methodwas introducedby Tezduyaret al.9.loto solve incompressibleflow problemsinvolving
free surfaces,two-liquid interfacesand fluid-structure and fluid-particle interactions.Later, similar
formulations were developedfor compressiblegas flowsI I and compressibleliquid flows.12
INT. J. NUMER. METH. FLUIDS, VOL 24: 1353-1369(1997)
(!:) 1997by John Wiley & Sons,Ltd.
SIMULATION
OF LARGE RAM-AIR
1355
PARACHln"ES
In this paper we discuss the steady glide aerodynamicsand inflation aerodynamicsof ram-air
parafoils. The computationsreported are performed on the Thinking Machines CM-5 and CRAY
T3D massivelyparallel computers.On the CM-5 we use an SIMD or data-parallelimplementation13
and on the T3D we use a message-passing
paradigmbasedon PVM.14
In thesecomputations,updating the finite element mesh as the spatial domain changesits shape
with time becomesa major issue.Thereare severalways of managingthis; the detaileddescriptionof
theseapproachestogetherwith their advantagesand disadvantagescan be found in Reference15. In
our studiesthe motion of eachfinite elementgrid point is prescribedexplicitly. This is accomplished
with no remeshing(i.e. without generatinga new set of nodesand elements).With this approachthe
connectivity of the mesh remains the same throughout the simulation. As a result, both the mesh
generationand parallelizationset-upcostsare reducedto a minimum. This is desirablefor practical
simulations with hundredsof time steps.
The organizationof this paper is as follows. The governing equationsare reviewed in Section 2.
The stabilized finite elementformulation is presentedin Section3. In Section4 we presentthe mesh
moving schemeused in our simulation of the inflation stage.Parallel implementation aspectsare
briefly coveredin Section5. The steadyglide and inflation simulationscompriseSection6. The flight
mechanicssimulationsare presentedin Section 7.
2. GOVERNING EQUATIONS
Let.Qt C ~nodand (0, 7) be the spatialand temporaldomainsrespectively,wherensdis the numberof
spacedimensions,
andlet r t denotethe boundaryof .Qt.The subscriptt implies the time dependence
of the spatial domain. The spatial and temporal coordinatesare denotedby x = (x,y,z) E .Qt and
t E (0, T). The governingequationsfor the flow field computationsare the Navier-Stokesequations
of incompressibleflows,
( au
P -;Ji+uoVu+f
-v . (J"= 0 on Q,
(1)
v . u = 0 on0,
(2)
wherep is the (constant)densityand u is the velocity vector.Here f is the externalforce consistingof
gravity. For the Newtonian fluids under considerationthe stresstensor for a fluid with dynamic
viscosity Jl is defined as
a(o,p)
= -pI
+ 2Jl£(0),
(3)
wherep is the mechanicalpressureand £(0) is the strain rate tensorgiven by
E(U)
= ![VU + (VU)T
(4)
Both Dirichlet- and Neumann-typeboundaryconditions are accountedfor, representedas
u=g
on (r')g'
n°O'=h
on (r,)1t,
where (r,)g and (r')h are complementarysubsetsof the boundary
velocity is specified as
O(X,0) = 00 on '>.0.
(5)
The initial condition on the
(6)
where 00 is divergence-free.
(!:) 1997 by John Wiley & Sons,Ltd.
INT. J. NUMER. METH. I'LUIDS. VOL 24: 1353-1369(1997)
1356
V. KALRO ET AL
The parafoil is treatedas a solid body with known geometric time variation. The Navier-Stokes
equations are coupled together with Newton's laws of motiort for the parafoil system. Purely
translationalmotion is considered.Theseequationsare
= (W /g)a,
F+ W
(7)
whereF is the aerodynamicforce acting on the parafoil and W is the gravitationalforce acting on the
parafoiljpayload system.Here a is the linear accelerationof the masscentreof the parafoiljpayload
system.
3. DEFORMING SPATIAL DOMAIN/STABILIZED
FORMULATION
SPACE-TIME
(DSD/SST)
In order to construct the finite element function spaces for the space-time method, we partition the
time interval (0, 7) into subintervals 1/1= (1/1'1/1+1),where 1/1and 1/1+1belong to an ordered series of
time levels 0 = 10 <
I( <
."
< IN
= T. Let.Q/I=.QI andr /I = r, ' We definethe space-timeslab
as the domain enclosed by the surfaces .Q/I'.Q/I+1a"ndP /I' where P/I is the surface described by the
boundaryr 1 asI traverses 1/1'As is the casewith r I' the surface P /I is decomposed into (P/I)g and (P /I)h
with respect to the type of boundary condition (Dirichlet or Neumann) being imposed. For each
space-time slab we define the corresponding finite element function spaces (9":)/1' ("1/':)/1'(9";)/1 and
("1/';)/1' Over the element domain this space is formed by using first-order polynomials in space and
time. Globally the interpolation functions are continuous in space but discontinuous in time.
The stabilized space-time formulation for deforming domains is then written as follows: given
(Uh);;, find uh E (9":)/1 and ph E (9";)/1 such that Vwh E ("1/':)/1and Vqh E ("1/';)/1
Q/I
J
Wh.
P
(~+
O.
uh
. Vuh+ f ) d.Q + J.
E(wh):a(ph, uh)dQ + J.
Q.
+
/1.1 J.
L
e=1
Q:
I
-tL(w,
P
q)
qhV' uhdQ
Q.
. [L(u,p) + pf]dQ + L/1.1 J.
c5V
.whpV. uhdQ
.
e=1 Q..
+ J (wh):' p[(Uh):- (Uh);;] d.Q = J
O.
Wh
. hhdP.
(8)
(P.)A
This processis appliedsequentiallyto all the space-timeslabsQ. , Q2
formulations given by equation(8), the following notation is used:
QN-lo In the variational
(9)
Jim u(tn
£-.0
JQ"
:i:: 8).
(10)
)dQ = J. J
I. Q.
)d.Qdt,
(II)
.)dP=
)drdt.
(12)
p.
INT. J. NUMER. METH. FLUIDS, VOL 24: 1353-1369(1997)
(.
«;) 1997 by John Wiley & Sons. Ltd.
SIMULATION
OF LARGE RAM-AIR
PARACHUTES
The computationsstart with
(Oh)O = 00.
Remarks
1. In the variational formulation given by equation(8), the first three termsand the right-handside
constitute the Galerkin formulation of the problem.
2. The first seriesof element-level integrals in equation (8) consistsof the least squaresterms
basedon the momentumequation.Here T is defined as
(~lIuhll)
T=
+(~)2- -1/2
2
h
where h is the elementlength and v = Jl/p.
3. The secondseriesof element-levelintegrals is addedto the formulation for numerical stability
at high Reynoldsnumbers.Theseare the least squaresterms basedon the continuity equation.
The coefficient is defined as
15
() 5)
where
z=
Reu/3, Reu~ 3,
Reu> 3
1,
and Reuis the cell Reynoldsnumber.
4. Both stabilization terms are weighted residualsand thereforemaintain the consistencyof the
formulation.
5. The sixth term enforces,weakly, the continuity of the velocity field acrossthe space-timeslabs.
4. MESH MOVING SCHEMES
In our finite elementcomputationswe considertwo categoriesof meshmoving schemes:automatic
schemesand special schemes.
In automatic schemes the mesh displacementsare prescribed on the boundaries and the
displacementsin the interior are determinedby solving the equationsof elasticity for the domain.IS
This schemeis very generaland particularly suitable for unstructuredmeshes;however,it involves
the additional cost of solving for the node displacements.Furthermore,when the mesh becomes
excessivelydistorted, remeshingneedsto be undertaken.This involves generatinga new meshand
projecting the solution from the old meshto the new mesh.Projectionand meshgenerationare timeconsumingand posebottlenecksin the parallel implementation.
In specialmesh-movingschemes,which are normally designedfor specific problems,the motion
of each node is prescribedexplicitly. In our computationswe utilize such a schemetogetherwith a
specially designed algebraic mesh generator.In its initial state prior to inflation the parafoil is
assumedto have the shapeof a box. In its final state it is fully inflated. The mesh connectivity
betweenthe two statesremain unchanged.The time for inflation is estimatedfrom drop test data.The
inflation processis modelled as a smoothtransformationbetweenthe two states.As a result of this,
the mesh generatoris used to generatemeshescorrespondingto the two end statesonly, and at a
~ 1997 by John Wiley & Sons,Ltd.
INT. J. NUMER. METH. RUmS, VOL 24: 1353-1369(1997)
]358
V. KALRO ET AL
given instant during inflation the mesh is interpolated from these two states with no need for
remeshing.At this time the pitching motion of the parafoil is constrained.
It is important to note that the true deformation history would come out of the solution of a
complex fluid-structure interaction problem which we plan to considerin the future.
5. PARALLEL IMPLEMENTATION
We briefly describeherethe parallel implementationof the finite elementalgorithmson the Thinking
MachinesCM-5 and CRAY T3D supercomputers.
The finite element formulations describedin earlier sectionsgive rise to very large systemsof
couplednon-linearequationswhich requirethe useof iterative strategieswith updatetechniquessuch
as GMRES16for their solution. To further reduce the memory requirements,we use matrix-free
iterations and thus eliminate the needto store element-levelmatrices.
The bulk of the computing cost is taken up by two tasks.
I. Computationof element-levelquantities.
2. Communication of data acrossprocessorswhile forming the global equation systems.This
involves the data transfer modes Gather (global/node-+ local/element) and Scatter
(global/node +-local/element).
Of thesetwo, the first task is fully parallel in the sensethat all operationsfor each element are
entirely local to a processor.The secondstepcan be a bottleneckand thuscareful considerationhasto
be given to the use and distribution of data structures to maximize localityl7.18 and minimize
communication.
On the CM-5 we usea data-parallelprogrammingparadigm.The task of synchronizingprocessing
nodesis internal to the machine.The meshis partitionedand communicationtracesarecomputedand
stored through calls to routines in the ConnectionMachine Scientific Software Library (CMSSL).
The interestedreaderis referred to References7 and 19 for further details.
On the T3D we use a message-passing
paradigmbuilt upon the CRAY extensionof the Parallel
Virtual Machine (PVM) software. Here synchronizationbetweenprocessorsis realized explicitly
throughPVM barriersplacedaccordinglyin the code.Gatherand scatteroperationsare performedby
routines programmedto offer the samefunctionality availableon the CM-5.
6. FINITE ELEMENT SlMULA TIONS
Steadystate simulationsat Re = 107
All the parafoil simulationsreportedin this Sectionare carried out on the CRAY T3D massively
parallel supercomputer.The aerodynamiccharacteristicsof a parafoil are the crucial quantitieswhich
determineits performance.We have carried out steadystatesimulationsat various anglesof attack
(a).
The mesh for the parafoil, with a Clark- Y cross-section with a rounded leading edge, consists of
291,437 nodes and 279,888 hexahedral elements. The aspect ratio of the parafoil is 3.0 and the ratio
of line length to span 0.6. The x-axis is in the chordwise direction and the y-axis along the span.
Boundary conditions consist of no-slip conditions on the parafoil surface, specification of the velocity
at the inflow boundary, zero shear stress and zero normal velocities at the side boundaries and
traction-free conditions at the outflow boundary. In these simulations, b refers to the parafoil span and
c to the chord length.
INT. J. NUMER. METH. FLUIDS. VOL 24: 1353-1369(1997)
(!d 1997by Jobn Wiley & Sons.Ltd.
SIMULATION'
OF LARGE RAM-AIR
PARACHUTES
1359
A semidiscreteformulation coupledwith matrix-free iterationswas usedto obtain the solutions.At
every step, 1,129,248couplednon-linear equationsare solved. Turbulenceis incorporatedinto the
simulationsusing a simplified version of the algebraicBaldwin-Lomax2omodel,
.Llt
= pflrol.
(16)
(17)
1= ICn[l - exp( -n+ /A+)].
where III is the eddy viscosity, w is the vorticity vector, " = 0.41 and A+ = 25.0 are constants,n is
the normal distance to the wall and n+ is the same distance expressedin wall units. In our
computationsthis distanceis measuredfrom the closestnode on the surface.
Figure 1 shows the chordwise pressure distribution on the parafoil surface at y I b
= 0.0 (midspan)
for various anglesof attack. The pressureprofile is similar to that of a 20 aerofoil with suction
present on the upper surface.Figure 2 shows the chordwise pressuredistribution on the parafoil
surfacecloseto the parafoil tip for variousanglesof attack.30 effectsare distinctly present,with the
suction decreasedowing to leakageof fluid around the tips from the lower to the upper surface.
Figures3-5 show the spanwisepressuredistribution on the parafoil surfaceat different chord lengths
for variousanglesof attack.The pressuredistribution is close to elliptical, with the effect of parafoil
bumps clearly prevalent at the leading edge; the distribution flattens out progressivelytowards the
trailing edge.
Figure 6 showsthe lift (C/) and drag (Cd) coefficients(which are basedon the freestreamvelocity
and the projectedareaof the parafoil) as functionsof the anglesof attack.To accountfor the inlet, a
factor21of 0.5hlc is added to the drag coefficient. Here h is the inlet height. A typical value of
hlc
= 0.1 is used. We compared these coefficients
with available experimental data on parafoils and
the calculatedvaluesare in the expectedrange.We could not carry out a closer comparisonbecause
of differencesin the shapeof our parafoil model and thosefor which data are available.Moreover,
while experimentaldatarevealthat mostparafoils stall at around8°-12°, our model doesnot indicate
this. We think that the reasonsfor this are as follows: first, our model hasa roundedleading edgeas
opposedto an inlet; second,the predictive capability of the turbulencemodel we use diminishesat
high anglesof attack in the regionsof flow separation;third, in our computationswe assumethat the
parachutesare not deformable,but this is not the case.Figure 7 showsthe lift-to-drag ratio (LID) asa
function of the angle of attack. The typical maximum LID for the ram-air parafoil alone is in the
0.8
0.6
++-
"_'O..
-8
0.4 ~
, ~
0.2
... '. .,.,
',_.::::=:::::::::~~:::-~~~:=!=.:::".,
aI
g' -0.2
~
-0.4
.."..
'
\. ~ ~,w- ,w
-o.~
-0.8
,.~..
... --::::.
..".
w'
~.~
..'-
u.c
Uo4
x/c
_0-
-:-
U.U
O.8~-C::
~
Figure I. Steady state simulations at Re = 107 for various angles of attack: chordwise pressure distribution on parafoil surface
at y/b = 0.00 (midspan)
,
<e>
1997by John Wiley & Sons,Ltd.
INT. J. NUMER. METH. RUIDS, VOL 24: 1353-1369(1997)
1360
V. KALRO ET AL.
O.A
.
.
.
0.6
"
..,
+_.
_10""
a
0.41
I
i
0.2
!
u.
GI
01
OS
(!J
~...
. ':::.~:~'.':..~'~...~..n
0
~'
.,
=::: ::::
~ '...
.. &..~'&.
-0.2
- -
r
... -+--
~:::-.:::.:-;=-;:.::: :::::::: ;:::.~:::.-:::. .
-0.41
-0."
..
Q~2
'
.
0.6
I
,.& ___0_."'
.A
A.#~.~
"'.A
'.~~.K
.
~
01,0.-02
-
+-.
"""'0", ...
e--
~.~~.A
,
...',~
""~A
po
0.4
E~ ~..-
.
!
I
0.2
.
Q~8
= 107for variousanglesof attack:chordwisepressuredistribution on parafoil surface
aty/b = 0.45
1
0.8
Q~6
~
&~.."~"
'
Figure 2. Steadystatesimulationsat Re
Q~4 x/c
~-
-0.8
0
..~
.
-0.4
-0.6
-0.8'
.. .., }."'a)
.A
. &...
.-..I~.
"
-0.4
Figure 3" Steady state simulations at Re
-0.2
y7b
~. J .
.
.
~.A'.-~
.~ I
-0.2
(G
.;-.-
8,
(!)
~
-
-"-
-","
u.~
U.4
= 107 for various angles of attack: spanwise pressure distribution
x/c
= 0.02
on parafoil surface at
0.8
~ -~_.
_do.
oj."""
""...IO~.
oa
0.4
~ 0.2
~
0
~-0.2
,
0"
..~...
~
It
++~~-~~.+~~~~~~.++~~~~+++~~~-
+~~~-~.
l :::::::~3i
~
~~- ~:::::::
8. ,
..
-0.4
~., ,.~
..
+
i
~
.~--~~~
-..
8~
, , , , I , """"~;;
::;:::;:::
\
-..,
~.
"
"..,
~-~++~_.~..+..~~-~~~~
,
"0".
~+~ ""'~n"
.-
..-~.w
.
-0.6
-0.6
I
~
.
.u...
Figure 4. Steady state simulations
at Re
=
107 for
I
x-
~
-u.~
y'1b
-'-
0.2
.,
0.4
angles of attack: spanwise pressure dishibution on parafoil surface at
various
x/c
INT. J. NUMER. METH. FLUIDS, VOL 24: 13S~1369 (1997)
= 0.25
<9 1997by John Wiley & Sons,Lid.
Plate I. Steady-state simulations at Re=IO7. at IX =2": pressure distribution on the parafoil surface
Plate 3. Steady-stale simulations al Re=107, at a =10': pressure distribution on the parafoil surface
Plate 4. Transfonnation of the box to a parafoil: pressure distribution on the parafoil surface during the
Ir"n.fnrm"l;nn
SIMULATION
Figure 5. Steady state simulations at Re
OF LARGE RAM-AIR
PARACHUTES
= 107 for various angles of attack: spanwise pressure distribution
x/c
= 0.50
1361
on parafoil surface at
range 3.0-5.0. This is somewhathigher than obtainablefor typical parafoil systems;however,our
model doesnot include line and payload drag.
To establishthe convergenceof our results,at an angleof attackof 6°, the pressuredistribution on
the parafoil surface is comparedwith that obtained on a mesh with 594,587 nodes and 575,968
elements.Figures8 and9 indicategood agreementbetweenthe solutionsobtainedon the two meshes.
Plates 1-3 show the pressuredistribution on the parafoil surfacefor various anglesof attack.
Transfonnationof the Box to a Parafoil
The inflation of a ram-air inflated gliding parachutetakesplace in three stages.4During the first
stagethe canopy expandswith little cell inflation. The canopy then pitches forwards and air rushes
into the separatecells of the gliding wing, causingthem to inflate. As the cells inflate, the parachute
begins to take on the shapeof an aerofoil, causing lift to be producedwhile drag decreases.The
parachutethen transitsinto equilibrium glide. Sincewe do not take into accountthe pitching motion
in the computationsreportedhere. only part of the entire processis simulated.
Figure 6. Steady state simulations at Re
~ 1997by John Wiley & Sons,Ltd.
= 107: drag and lift
coefficients as functions of angle of attack
INT. J. NUMER. METH. FLUIDS. VOL 24: 1353-1369(1991)
1362
V. KALRO ET AL
3.8
3.6
3.4
3.2
2.8
2.6
10
.-
alpha
Figure 7. Steady state simulations at Re
= 107: lift/drag
0.8 I'
0.61
(L/D) ratio as function of angle of attack
.
.
NES~I::::
M~H;
0.4
,i
!
!
0.2
0 ,~,,'I"'
~:..-
...
Q)
0)
~
,
-0.2
"
~,
'
I'.~~.
"""~~~~~:;:;~~:;4J:m:;~~$:$'#/~
-~
.
~I
~
-0.41
I
-0.6
-0.8~o'.4
:0'.2
=
y~
0:2
O:4J
Figure 9. Steady state simulations at Re
107 for a = 6°; spanwise pressure distribution on parafoil
xjc = 0.25. Comparison solutions on mesh I (291,437 nodes) and mesh 2 (594,587 nodes)
INT. J. NUMER. METH. FLUIDS, VOL 14: 13SJ-1369(1997)
surface at
«;) 1997 by John Wiley & Sons.Ltd.
SIMULATION
OF LARGE RAM-AIR
PARACHUTES
1363
All the parafoil simulationsreportedin this caseare carried out on the Thinking MachinesCM-5
massively parallel supercomputer.In these computationsthe parafoil is allowed to fall under the
influenceof gravity, inflate and tend towardsa steadygliding state.Here a partially inflated box with
dimensionsof chord x span x thickness= 48.0 x 33.4 x 12.0 ft3 transformsto a gliding parafoil at
a prescribedrate. The box expandsby factors of 1.5 and 6.5 along the chord and spanrespectively
and the cross-sectionbecomesthe sameas that of an NACA 0025 aerofoil. Figure 10 shows the
surfacemeshfor the transformationof the box to a parafoil. The meshin the middle is interpolated
from the two others using the meshmoving schemedescribedearlier.
The finite elementmeshusedin this simulation consistsof 170,950nodesand 161,856hexahedral
elements.The boundaryconditionsconsistof specifyingthe freestreamvelocity (which is zero) at the
inflow boundary, zero normal velocity and zero shear stressat the side boundaries,traction-free
conditionsat the outflow boundaryand no-slip conditionson the parafoil surface.At every time step,
1,304,606coupled non-linear equationsare solved.
The computation starts at t = 0.0 s, which correspondsto the steady state solution at the initial
configuration of the box at 10° angle of attack and with a velocity of 112 ft S-I. At t = 2.0 s the
Figure 10. Surface mesh for transformation of box to parafoil
(!:) 1997by John Wiley & Sons,Ltd.
INT. J. NUMER. METH. FLUIDS. VOL 24: 1353-136911997)
~
1364
V. KALRO ET AL
transformation ends, but the computation continues until t = 3.5 s. The parafoiljpayloadsystem
massfor this caseis 22,000 lb.
Figures II and 12 show the time historiesof the projection areaand velocity respectively.Initially
the box has very little lift and gains speedowing to gravity. As it expandsrapidly at high drop
velocities, very large aerodynamicforces are generated,leading to the peaks in Figure 13. We
18000I
.
.
.
.
.
.
I
16000
14000
-
12000
-~10000
: 8000
6000
.4000
2000
0
3.5
Figure 11.
.
.
.
.
.
-u -w --
Maonitude
120
100
velocltv
.
of
140
.
=:~
.
"'Vi
~ 80
.~
'6 60
~
40 I
1.5
2
~"
-.-
3
~~
---
TIme (5)
Figure 12. Transformation of box to parafoil: time history of parafoil velocity
80000
70000
60000
Drl\g
Magnitude of force
Lift
..
50000
.40000
1
1.5
2
2.5
3
3.5
Tlme(s)
Figure 13. Transfonnationof box to parafoil: time history of forces acting on parafoil
INT. J. NUMER.
METH. FLUIDS, VOL 24: 1353-1369 (1997)
(!:) 1997by John Wiley & Sons,Ltd.
SIMULATION
OF LARGE RAM-AIR
PARACHUTES
observethat the parafoil attainsan almost steady4° angleof attackas indicatedby Figure 14. Plate4
showsthe pressuredistribution on the parafoil surfaceduring the transformation.
~-~
dt - 2
(19)
The trajectory and openingforce can be determinedby solving theseequationson a computer.The
primary task is to model correctly the evolution of the lift and drag areasas functionsof time. The lift
and drag areaswere extractedfrom data on drop tests9-309 and 9-310 of ram-air inflated personnel
parachutesand are shownin Figure 15.During the initial inflation of the canopythe airflow is nearly
perpendicularto the chord of the canopyand the aerofoil is at a 90° angle of attack. This causesthe
drag coefficient of the parachuteto be much higher than during gliding flight. In addition, the airflow
is perpendicularto the inlet openingsof the cells. Thus thesecells do not inflate and little lift is
produced.
During canopyinflation the simulationusesa parabolicgrowth curve for the drag areaasa function
of time. By matching the simulation output to measureddata, the drag area was found to have a
maximumvalue of 0.80Soduring the period of canopyinflation, whereSois the canopyarea.The lift
area was assumedto be zero during this time.
The time of inflation for the canopy was determinedusing the relation
KI = (rlV'Vc:
0
~ 1997 by John Wiley & Sons.Ltd.
(21)
INT. J. NUMER. METH. FLUIDS. VOL 24: 1353-1369(1997)
1366
V. KALRO ET AL
Figure 15. Flight mechanics analysis of drop 9-309: lift and drag areas as functions of time
whereDo = (So4/7t)0'5,
If! is the time for the drag areato grow from zero to its full value (0.8So)and
Vs is the parachutevelocity at line stretch.By matching the simulation output to the measureddata
for personnel-typegliding parachuteswith slider reefing,K\ was determinedto have a value of 18.0.
Near the end of inflation the canopybeginsto pitch forwards,causingthe cells to inflate. As they
inflate, the drag coefficient drops and lift beginsto be produced.The time betweenthe beginning of
cell inflation and the beginning of the glide stage, lc2' is determinedsimilarly to lc\' except that
K( = 2.5 and Vs is the velocity at the beginning of cell inflation. Once again the simulation usesa
paraboliccurve for the lift areagrowth and drag areareductionas a function of time. During the final
stageof inflation the drag and lift areasare assumedto be constant,with Cdo = 0.21 and Clo = 0.58.
The simulation was applied to an MC-4 personnel-type parachute with the following
characteristics:
So = 370.0ft'-.
b = 28.5 ft,
c = 13.0 ft.
The rigged masswas 360 lb. A great deal of detailedinformation was availableon a numberof tests
for this parachute.The simulationis shownto give good resultswhen comparedwith measureddata.
Figures 16 and 17 show comparisonsbetweencalculatedforce and measuredforce on the parachute
for two separatedrop tests.The simulation predicts a peak force almost equal to the measuredpeak
force in both cases,but it showsthe peak force occurring before the actual measuredforce.
INT. J. NUMER. METH. FLUIDS, VOL 24: 1353-1369(1997)
~ 1997 by John Wiley & Sons,Ltd.
SIMULATION
OF LARGE RAM-AIR
PARACHUTES
3O(J}
1367
'-
25M
2OOC
~§
ISOC
..0
100"
s~
-~I-.
2S.S
26
.
26.5
.
27
.
21.5
28
28.5
29
Time(s..x.Ids)
Figure 17. Flight mechanics analysis of drop 9-310: comparison between computed and measured forces
Figure 18 shows the simulatedparachutevelocity as a function of time. It can be seenthat the
velocity drops quickly to its steadystategliding value. The flight path angle as a function of time is
shown in Figure 19. Initially the payloadis falling vertically, but as the parachutedeploys and starts
to glide, the initial path angle decreases.Figure 20 shows the altitude as a function of range; this
clearly shows that the parachuteis entering into its steady state gliding mode. This method was
Figure 18. Flight mechanics analysis of drop 9-309: parachute velocity as function of time
(!::) 1997 by John Wiley & Sons, Ltd.
INT. J. NUMER. MEm. FLUIDS. VOL 24: 1353-13690997)
1368
V. KALRO ET AI..
g
i
.'~Jo
20
40
60
80
100
120
140
160
180
200
RAna.(II)
Figure
20. Hight
mechanics
analysis
of drop 9-309:
parachute
altitude
as function
of range
applied to inflation of a large gliding parachu~ewith sequential reefing. After considerable
manipulation of the lift and drag functions it was possible to obtain opening forces which were
similar to experimentaldata.However,this approachrequiresaccessto flight test data and would not
be useful for analysis of a design for which no data are available. We are currently working on
improving our finite elementmodel to use it to predict accuratedrag and lift areashistoriesfor such
situations.
8. CONCLUDING REMARKS
We demonstratedthe useof two approachesfor the simulationof parachutedynamics.One approach
employedstate-of-the-artfinite element formulations and massively parallel computing technology
for numerical simulation of flow around a parafoil. Computed lift and drag coefficients and L/ D
ratios for steady glide were in reasonable agreement with experimental values. Preliminary
simulationsof the inflation stageswere made,with the assumptionthat the time history of the parafoil
surfacethrough the transformationis known. The actual evolution of the surface would involve a
complex fluid-structure interactionproblem and this will be the direction of our future research.The
secondapproachincorporatedlift and drag time historiesfrom flight data.Thesehistorieswere input
to the flight mechanicsequations.The useof this methodis limited to parachutessimilar to thosefor
which data are available. As our computationalmodel evolves, we hope to use a hybrid technique
where flight data could be obtainedfrom numericalsimulations,thus reducingexpensivedrop tests.
ACKNOWLEDGEMENTS
This researchwas sponsoredby ARO under grant DAAH04-93-G-O514,by ARPA under NIST
contract 60NANB2DI 272, by NASA-HSC under grant NAG 9-449 and by the Army High
PerformanceComputing ResearchCenterunder the auspicesof the Departmentof the Army, Army
Research Laboratory co-operative agreement number DAAH04-95-2-0003/contract number
DAAH04-95-C-OOO8,
the content of which does not necessarilyreflect the position or the policy
of the government,and no official endorsementshouldbe inferred. CRAY C90 time was provided in
part by the University of MinnesotaSupercomputerInstitute.
REFERENCES
I. W. Wailes, 'Development testing of large ram air inflated wings', AIAA Paper 93-1204, 1993.
2. W. L. Garrard, T. E. Tezduyar, S. K. Aliabadi, V. Kalro, J. Luker and S. Mittal, 'Inflation analysis of ram air inflated
gliding parachutes', AIM Paper 95-1565, 1995.
INT. J. NUMER. METH. FLUIDS, VOL 24: 1353-1369(1997)
(!::) 1997 by John Wiley & Sons, Ltd.
SIMULATION
OF LARGE RAM-AIR
PARACHUTES
1369
3. S. K. Aliabadi, W. L. Garrard, V. Kalro, S. Mitla!, T. E. Tezduyar and K. R. Stein, 'Parallel finite element computations of
the dynamics of large ram air parachutes', AIM Paper 95-1581, 1995.
4. J. S. Lingard, 'A semi-empirical theory to predict the load-time history of an inflating parachute', AIM Paper 84-0814,
1984.
5. J. S. Lingard, 'Unsteady aerodynamics', in University of Minnesota Parachute Systems Technology Short Courses, League
City, TX, 1994.
6. W. Garrard, 'Application of inflation theories to preliminary force and stress analysis', AIM Paper 91-0862, 1991.
7. T. Tezduyar, S. Aliabadi, M. Behr, A. Johnson and S. Mittal, 'Parallel finite-element computation of 3D flows', IEEE
Comput., 26(10), 27-36 (1993).
8. T. E. Tezduyar, S. K. Aliabadi, M. Behr and S. Mittal, 'Massively parallel finite element simulation of compressible and
incompressible flows', Comput. Methods Appl. Mech. Eng., 119, 157-177 (1994).
9. T. E. Tezduyar, M. Behr and J. Liou, 'A new strategy for finite element computations involving moving boundaries and
interfaces-the
deforrning-spatial-domain/space-time
procedure: I. The concepts and the preliminary tests', Comput.
Methods Appl. Mech. Eng., 94, 339-351 (1992).
10. T. E. Tezduyar, M. Behr, S. Mittal and J. Liou, 'A new strategy for finite element computations involving moving
boundaries and interfaces-the deforrning-spatial-domaon/space-time
procedure: II. Computation of free-surface flows,
two-liquid flows, and flows with drifting cylinders', Comput. Methods Appl. Mech. Eng., 94, 353-371 (1992).
11. S. K. Aliabadi and T. E. Tezduyar, 'Space-time finite element computation of compressible flows involving moving
boundaries and interfaces', Comput. Methods Appl. Mech. Eng., 107, 209-224 (1993).
12. G. P. Wren, S. E. Ray, S. K. Aliabadi and T. E. Tezduyar, 'Space-time finite element computation of compressible flows
between moving components', Int. j. numer. methods fluids, 21, 981-991 (1995).
13. M. Behr, A. Johnson, J. Kennedy, S. Mittal and T. E. Tezduyar, 'Computation of incompressible flows with implicit finite
element implementations on the Connection Machine', Comput. Methods Appl. Mech. Eng., 108,99-118 (1993).
14. A. Geist, et al., PVM: Parallel Virtual Machine. A User's Guide for Networked Parallel Computing, MIT Press,
Cambridge, MA, 1994.
15. A. A. Johnson and T. E. Tezduyar, 'Mesh update strategies in parallel finite element computations of flow problems with
moving boundaries and interfaces', Comput. Methods Appl. Mech. Eng., 119, 73-94 (1994).
16. Y. Saad and M. Schultz, 'GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems',
SIAM J. Sci. Stat. Comput., 7, 856--869 (1986).
17. A. Pothen, H. D. Simon and L. Wang, 'Spectral nested dissection', Tech. Rep. RNR-92-003, NASA Ames Research Center,
Moffett Field, CA, 1992.
18. Z. Johan, 'Data parallel finite element techniques for large-scale computational fluid dynamics', Ph.D. Thesis, Department
of Mechanical Engineering, Stanford University, 1992.
19. V. Kalro and T. Tezduyar, 'Parallel finite element computation of 3D incompressible flows on MPPs', in W. G. Habashi
(ed.), Solution Techniques for Large-Scale CFD Problems, Wiley, New York, 1995, pp. 59-81.
20. B. Baldwin and H. Lomax, 'Thin layer approximation and algebraic turbulence model for separated turbulent flows', AIM
Paper 78-257, 1978.
21. J. S. Lingard, 'Ram-air parachutedesign', AIM Paper, 1995.
~ 1997by John Wiley & Sons,Ltd.
INT. J. NUMER. METH. fLUIDS, VOL 24: 1353-1369(1997)