NUMERICAL MODELING OF AERODYNAMICS OF AIRFOILS OF MICRO AIR VEHICLES IN ...

156 Pages·2008·3.49 MB·English

NUMERICALMODELINGOFAERODYNAMICSOFAIRFOILSOFMICROAIR
VEHICLESINGUSTYENVIRONMENT
ADissertation
Presentedto
TheGraduateFacultyofTheUniversityofAkron
InPartialFulfillment
oftheRequirementsfortheDegree
DoctorofPhilosophy

HarishGopalan
December,2008
NUMERICALMODELINGOFAERODYNAMICSOFAIRFOILSOFMICROAIR
VEHICLESINGUSTYENVIRONMENT
HarishGopalan
Dissertation
Approved: Accepted:
Advisor DepartmentChair
Dr. AlexPovitsky Dr. CelalBatur
CommitteeMember DeanoftheCollege
Dr. GrahamKelly Dr. GeorgeK.Haritos
CommitteeMember DeanoftheGraduateSchool
Dr. ScottSawyer Dr. GeorgeR.Newkome
CommitteeMember Date
Dr. ErnianPan
CommitteeMember
Dr. KevinKreider
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ABSTRACT
Thesuperiorflightcharacteristicsexhibitedbybirdsandinsectscanbetakenasaprototype
of the most perfect form of flying machine ever created. The design ofMicro Air Vehicles
(MAV) which tries mimic the flight of birds and insects has generated a great deal of
interest as the MAVs can be utilized for a number of commercial and military operations
which is usually not easily accessible by manned motion. The size and speed of operation
of a MAV results in low Reynolds number flight, way below the flying conditions of a
conventionalaircraft. Theinsensitivitytowindshearandgustisoneoftherequiredfactors
to be considered in the design of airfoil for MAVs. The stability of flight under wind
shear is successfully accomplished in the flight of birds and insects, through the flapping
motion of their wings. Numerous studies which attempt to model the flapping motion of
the birds and insects have neglected the effect of wind gust on the stability of the motion.
Also sudden change in flight conditions makes it important to have the ability to have an
instantaneouschangeoftheliftforcewithoutdisturbingthestabilityoftheMAV.
In the current study, two dimensional rigid airfoil, undergoing flapping motion
is studied numerically using a compressible Navier-Stokes solver discretized using high-
order finite difference schemes. The high-order schemes in space and in time are needed
to keep the numerical solution economic in terms of computer resources and to prevent
vortices from smearing. The numerical grid required for the computations are generated
using an inverse panel method for the streamfunction and potential function. This
grid generating algorithm allows the creation of single-block orthogonal H-grids with
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ease of clustering anywhere in the domain and the easy resolution of boundary layers.
The developed numerical algorithm has been validated successfully against benchmark
problemsincomputationalaeroacoustics(CAA),andunsteadyviscousflows.
Thenumericalresultsforpure-plungeandpure-pitchingmotionofSD7003airfoil
are compared with the particle image velocimetry data of Michael Ol1 by plotting the
contoursofstreamwisevelocityandvorticityandalsobyobservingthewakeprofileofthe
streamwise velocity. A very good agreement in the location of the vortices was observed
betweenthenumericalandexperimentalresults. Alsothenumericaltrackingofstreaklines
wascomparedwiththedyeinjectionexperimentsandexcellentagreementinthehorizontal
andverticallocationsofthevortexcoreswasobserved. Theimportanceofusingtheangle
of attack to match the wake structures and lift forces of airfoils in pure-pitch and pure-
plunge was investigated and it was found that matching the plunging amplitude with the
maximum displacement of the leading edge provides a closer match in the observed wake
structuresandcoefficientoflift.
Next, the average coefficient of list of an airfoil in pure-pitch was studied and it
was found that the pitching about the leading edge produced the maximum value. Two
difference methods of enhancements were considered: (i) axis of rotation, and (ii) moving
airfoil, as possible ways to enhance the average coefficient of lift for an airfoil pitching
about its leading edge. The first case produced two times increase and the second case
produced almost four times increase in the average coefficient of lift respectively. Hence
these two kinds of motion can be used for lift enhancement to overcome sudden changes
intheflightconditions.
Finally the effect of a sinusoidal gust on an airfoil in pure-pitch and pure-
plunge motion was examined. The pitching motion showed a much lesser drop in the
average coefficient of lift compared to the plunging motion, suggesting its effectiveness to
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overcome disturbances in the freestream. The plunging motion on the other hand can be
employedforcasesthatrequirethesuppressionoftheoscillationintheliftcoefficient.
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ACKNOWLEDGEMENTS
Firstandforemost,IwouldliketoacknowledgemyadvisorDr. Povitskyforgivingmethis
opportunity to work on the flapping airfoil project. He has been a source of knowledge,
inspiration,andguidancetome. Ihaveimmenselyenjoyedworkingwithhim. Ialsowould
like to thank Dr. Kelly, Dr. Sawyer, Dr. Pan and Dr. Kreider, for agreeing to serve on my
thesis committee in spite of their busy schedules. A special thanks to all my colleagues
(Kedar, Shinliu and Nathan ) and my roommate (Antonio) for providing encouragement
during the course of the research and for their continuing support. My family in India
and my brother in Baltimore for their endless love. Dr. Sri, for emphasizing the need for
effectivecommunicationandhisconstantencouragement. DrKreider,forforcingachange
inmycodingstyleinhisnumericalmethodsforpartialdifferentialequationscourse.
This work was sponsored in part by the graduate teaching fellowship of the
department of mechanical engineering at the University of Akron. This research was also
supportedbythefundingprovidedbytheAirforceofficeofscientificresearchatWPAFB,
Dayton, Ohio under the guidance of Dr.Michael V. Ol. A portion of the computing was
performedontheIBM-OpteronclusterattheOhiosupercomputingcenter.
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NOMENCLATURE
E Totalenergy
t
E,F Fluxes(inviscidandviscousincludes)
U Solutionvector
B Numericalparameterforclusteringalongφdirection
dt Timestep
f Frequencyofoscillation
h Plungingamplitude
k Reducedfrequency
p Pressure;acousticpressure
Re Reynoldsnumber
S Sourceterm
s,P,Q Numericalparametersforclusteringalongψ direction
St Strouhalnumber
u Cartesianx-componentofvelocity
v Cartesiany-componentofvelocity
α Maximumangleofpitching
0
α Spectralfunction
f
α Geometricangleofattack
g
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¯
C Averagecoefficientoflift
L
∂φ, ∂φ Gridmetrics
∂x ∂y
∂ψ, ∂ψ Gridmetrics
∂x ∂y
ω Angularfrequency
φ Phaseofmotion(Chapter5,6)
φ Phaseofmotion(Chapter5,6)
φ,ψ Potentialfunctionandstreamfunction;numericaldomain
ρ Density
σ Vortexstrengthinpanelmethods
i
τ Clusteringparameteralongφdirection
τ ,τ ,τ Componentsofshearstress
xx xy yy
C Coefficientoflift
L
C Peakamplitudeofliftcoefficient
Lpeak
M FreestreamMachnumber
∞
U Freestreamvelocity
∞
u ,u Contravariantvelocitycomponents
φ ψ
x Surgingamplitude
0
x ,y Centerofpulse
c c
x ,y Gridvelocities
t t
x(t),y(t) Kinematicmotionofairfoil
x,y Physicaldomain
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TABLEOFCONTENTS
Page
LISTOFTABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv
LISTOFFIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi
CHAPTER
I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Organizationofthethesis . . . . . . . . . . . . . . . . . . . . . . . . . 5
II. LITERATUREREVIEW . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 Historyofvehicleswithflappingairfoils . . . . . . . . . . . . . . . . . 7
2.2 Kinematicofflappingairfoils . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Aerodynamicsofflappingairfoils . . . . . . . . . . . . . . . . . . . . 10
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2.4 High-orderschemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.5 Gridgeneration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
III. DEVELOPMENT OF MATHEMATICAL MODEL, NUMERICAL
DISCRETIZATIONANDORTHOGONALGRIDGENERATION . . . . 21
3.1 GoverningEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 NumericalMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2.1 Implicitcompactspatialdifferencing . . . . . . . . . . . . . . . 23
3.2.2 ImplicitPadeTypeFilters . . . . . . . . . . . . . . . . . . . . . 24
3.2.3 ExplicitLowStorageRunga-Kuttatimeintegration . . . . . . . 26
3.2.4 BoundaryConditions . . . . . . . . . . . . . . . . . . . . . . . 26
3.2.5 ParticleTracing . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 GridGeneration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3.1 AnalyticalMethod . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3.2 GridClustering . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.3.3 NumericalMethod . . . . . . . . . . . . . . . . . . . . . . . . 34
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