Proceedings of Proceedings of the ASME 2013 Fluids Engineering Summer ConferenceFEDSM2013

July 7-11, 2013, Incline Village, Nevada

FEDSM2013-16540

IMPLICIT LES SIMULATIONS OF A FLAPPING WING IN FORWARD FLIGHT

Raymond E. GordnierAir Force Research Laboratory

Aerospace Systems DirectorateWright-Patterson AFB, OH 45433-7512Email: raymond.gordnier@wpafb.af.mil

Luciano DemasiDepartment of Aerospace Engineering

San Diego State UniversitySan Diego, California 92182

Email:ldemsi@mail.sdsu.edu

ABSTRACTComputations of an aspect ratio 3.5 flat plate wing in

flapping forward flight are performed. A high-order im-plicit LES approach is employed to compute the mixed lami-nar/transitional/turbulent flowfields present for the low Reynoldsnumber flows associated with micro air vehicles. The ILES ap-proach is implemented by exploiting the properties of a well val-idated, robust, sixth-order Navier-Stokes solver. The analyzedkinematics are a flapping motion described by an anti-clockwise8 cycle. A Reynolds number based on the freestream velocity of1250 is prescribed. A detailed description of the dynamic vortexsystem engendered by the unsteady flapping motion is given andrelated to the development of lift and thrust during the flappingcycle. Effective angle of attack, which results from the wing mo-tion, and its interplay with the aerodynamic angle of attack playa key role in determining the flow structure and forces produced.

IntroductionInsects present excellent flight performance [1, 2] and are

the ideal candidates for bio-inspired flapping unmanned aerialsystems. Hovering and forward flight of insects require a highflapping frequency which depends on the insect’s weight. Theinsect must be able [3] to accelerate the wing in one direction(downstroke), decelerate it and reverse the motion (supination),perform the upstroke, decelerate the wing again and reverse themotion for the next downstroke (pronation). These accelerationsand decelerations imply an increase/decrease of the kinetic en-ergy of the wing. An effective design of flapping unmannedaerial systems will try to harness the advantageous biological fea-

tures required for efficient flight with focus on the maximizationof the payload and minimization of the power required to flapthe wings. Successful development of these biomimetic flappingunmanned aerial systems will require significant advancementsin the fundamental understanding of the unsteady aerodynam-ics of low Reynolds number fliers. This computational approachhas been successfully applied to a variety of problems related toMAV flight [4–9].

Conventional simplified analytical techniques and empiricaldesign methods, although attractive for their efficiency, may havelimited applicability for these complicated, multidisciplinary de-sign problems as they do not capture critical flow features. Inthe present work a high-order implicit LES approach [10] isemployed to compute the mixed laminar/transitional/turbulentflowfields present for the low Reynolds number flows associ-ated with micro air vehicles. This ILES approach exploits theproperties of a well validated, robust, sixth-order Navier-Stokessolver [11–13].

The focus of this paper will be the simulation and analysisof aeroscience issues associated with an aspect ratio 3.5 wingin forward flapping flight. The particular flapping kinematicschosen are an anti-clockwise 8 flapping cycle first explored byViswanath and Tafti [14]. An initial investigation of the impactof the structural properties for an equivalent flexible wing waspresented by Demasi et al [15]. A detailed analysis of the flowstructures generated by the flapping motion and their relation tothe forces produced is given.

1

Wing Description and KinematicsForward flight is assumed to take place where a freestream

velocity V∞ (see Figure 1) is assumed constant in magnitude anddirection. The stroke plane χ is assumed to be on a vertical planeperpendicular to the horizontal one, as presented in Figure 1. Theflapping motion is described by an anti-clockwise eight flappingcycle. The “O” portions (identified by the second half of theupstroke and the first half of the downstroke or identified by thesecond half of the downstroke and the first half of the upstroke)of the “8” cycle are perfectly identical. The axes xS and yS are onthe stroke plane and zS is perpendicular to it. The axis yS is alsothe intersection between the stroke and horizontal planes.

FIGURE 1. Anti-clockwise 8 flapping cycle: case of vertical planestroke plane and freestream velocity directed along zS. In this case theyS− zS plane is coincident with the horizontal plane.

The wing computed is rectangular and perfectly planar (i.e.,no built-in twist or camber are present), Figure 2. It is possible to

FIGURE 2. Geometry of the rectangular wing.

demonstrate that the aerodynamic angle of attack is influenced byall Euler’s angles ϕ , γ , and ψ . These angles are useful to movefrom the wing-attached coordinate system (xw, yw, and zw) tothe stroke plane coordinate system (xS, yS, and zS) and viceversa.Figure 3 shows the positive convention. The parameters that need

FIGURE 3. Definition of Euler’s angles: flapping angle ϕ , deviationangle γ , and pitch angle ψ .

to be prescribed, for a given rectangular wing (see Figures 2 and3) are the flapping angle ϕ , the deviation angle γ and the pitchangle ψ .

Kinematic Laws Regulating the Euler AnglesThe flapping angle is assigned with the following law:

ϕ (t) =Φ

2· cos

(2π

tT

)(1)

where the stroke amplitude Φ = 60. The deviation angle, γ , isnegative over the first half of the downstroke. After the midpointof the downstroke the deviation angle is positive. At the begin-ning of the upstroke it is again negative and at the end of theupstroke it is positive. The law chosen to describe this behavioris:

γ (t) =−γ · sin(

4πtT

)(2)

where γ = 10. The kinematic law adopted for the pitch angle ψ

is:

ψ (t) =−ψP +ψ · sin(

2πtT

)(3)

where ψ = 32. ψP is the rotation angle at the pronation phase,and is specified to be 10. The kinematic laws are graphicallyportrayed in Figures 4-6.

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FIGURE 4. Anti-clockwise 8 flapping cycle: kinematic law of theflapping angle.

FIGURE 5. Anti-clockwise 8 flapping cycle: kinematic law of thedeviation angle.

Aerodynamic and Effective Angle of Attack

To understand the aerodynamics for the problem being con-sidered it will be helpful to define the concept of the aerodynamicangle of attack and the effective angle of attack. The aerody-namic angle of attack is defined in a plane perpendicular to thestroke plane, Figure 7. The equation for the line on the wing in

FIGURE 6. Anti-clockwise 8 flapping cycle: kinematic law of therotational motion ψ .

Figure 7 can be expressed as

xS =cosγ sinψ

cosψ cosφ − sinγ sinψ sinφzS

− Dcosψ sinφ + cosφ sinγ sinψ

cosψ cosφ − sinγ sinψ sinφ(4)

where D is the distance to the plane Ωa. The aerodynamic angleof attack can be evaluated from the slope of the line as follows:

tanαaeroa =

dxS

dzS=

cosγ sinψ

cosψ cosφ − sinγ sinψ sinφ(5)

To compute the effective angle of attack the velocity of thewing surface must first be calculated from the relation:

VVV flap = ωωω× r (6)

where

ωωω = ωxS iiiS +ωyS jjjS +ωzS kkkS

= (γ cosϕ− ψ cosγ sinϕ) iiiS +(γ sinϕ + ψ cosγ cosϕ) jjjS

+(ϕ + ψ sinγ)kkkS. (7)

The velocity of any point on the wing may then be defined as

VVV flap =(ωyS zS−ωzS yS

)iiiS +

(ωzS xS−ωxS zS

)jjjS

+(ωxS yS−ωyS xS

)kkkS. (8)

3

FIGURE 7. Aerodynamic angle of attack for a rectangular wing per-forming an anti-clockwise cycle

Using this expression the effective angle of attack may be definedas

αeffa = α

aeroa −ϑ a (9)

where

tanϑ a =−uflap

V∞−wflap(10)

Aerodynamic Governing EquationsThe governing equations solved are the three-dimensional,

compressible Navier-Stokes equations. These equations arecast in strong conservative form introducing a general time-dependent curvilinear coordinate transformation (x,y,z, t) →

(ξ ,η ,ζ ,τ). In vector notation, and employing non-dimensionalvariables, the equations are:

∂

∂τ

(~UJ

)+

∂ F∂ξ

+∂ G∂η

+∂ H∂ζ

=1

Re[∂ Fv

∂ξ+

∂ Gv

∂η+

∂ Hv

∂ζ] (11)

Here ~U = ρ,ρu,ρv,ρw,ρE denotes the solution vectorand J is the transformation Jacobian. The inviscid and viscousfluxes, F , G, H, Fv, Gv, Hv can be found, for instance, in Refer-ence 16. The system of equations is closed using the perfect gaslaw p = ρT/γM2

∞, Sutherland’s formula for viscosity, and the as-sumption of a constant Prandtl number, Pr = 0.72. In the expres-sions above, u,v,w are the Cartesian velocity components, ρ thedensity, p the pressure, and T the temperature. All flow variableshave been normalized by their respective freestream values ex-cept for pressure which has been nondimensionalized by ρ∞u2

∞.

Spatial DiscretizationA finite-difference approach is employed to discretize the

flow equations. For any scalar quantity, φ , such as a metric, fluxcomponent or flow variable, the spatial derivative φ ′ along a co-ordinate line in the transformed plane is obtained by solving thetridiagonal system:

αφ′i−1 +φ

′i +αφ

′i+1 = b

φi+2−φi−2

4+a

φi+1−φi−1

2(12)

where α = 13 , a= 14

9 and b= 19 . This choice of coefficients yields

at interior points the compact five-point, sixth-order algorithmof Lele [17]. At boundary points 1, 2, IL− 1 and IL, fourth-and fifth-order one-sided formulas are utilized which retain thetridiagonal form of the interior scheme [11, 18].

Compact-difference discretizations, like other centeredschemes, are non-dissipative and are therefore susceptible to nu-merical instabilities due to the growth of spurious high-frequencymodes. These difficulties originate from several sources includ-ing mesh non-uniformity, approximate boundary conditions andnonlinear flow features. In order to ensure long-term numeri-cal stability, while retaining the improved accuracy of the spa-tial compact discretization, a high-order implicit filtering tech-nique [13, 19] is incorporated. If a component of the solutionvector is denoted by φ , filtered values φ are obtained by solvingthe tridiagonal system,

α f φi−1 + φi +α f φi+1 = ΣNn=0

an

2(φi+n +φi−n) (13)

Equation 13 is based on templates proposed in Refs. 17and 20, and with proper choice of coefficients, provides a 2Nth-order formula on a 2N + 1 point stencil. The coefficients,

4

a0,a1, . . .aN , derived in terms of the single parameter α f usingTaylor- and Fourier-series analyses, are given in Ref. 11, alongwith detailed spectral filter responses. In the present study, aneighth-order filter operator with α f = 0.3 is applied at interiorpoints. For near-boundary points, the filtering strategies de-scribed in Refs. 13 and 12 are employed. Filtering is appliedto the conserved variables, and sequentially in each coordinatedirection.

Time IntegrationFor wall-bounded viscous flows, the stability constraint of

explicit time-marching schemes is too restrictive and the use ofan implicit approach becomes necessary. For this purpose, theimplicit approximately-factored scheme of Beam and Warming[21] is incorporated and augmented through the use of Newton-like subiterations in order to achieve second-order temporal andsixth-order spatial accuracy. In delta form, the scheme may bewritten as [

J−1p+1+φ i∆tsδξ

(∂ F p

∂U −1

Re∂ F p

v∂U

)]Jp+1×[

J−1p+1+φ i∆tsδη

(∂ Gp

∂U −1

Re∂ Gp

v∂U

)]Jp+1×[

J−1p+1+φ i∆tsδζ

(∂ H p

∂U −1

Re∂ H p

v∂U

)]∆U

=−φ i∆ts[J−1p+1 (1+φ)U p−(1+2φ)Un+φUn−1

∆t

−U p((

ξtJ

)ξ

+(

ηtJ

)η+(

ζtJ

)ζ

)p+1

+δξ

(F p− 1

Re F pv)+δη

(Gp− 1

Re Gpv)+

+δζ

(H p− 1

Re H pv)]

(14)

where

φi =

11+φ

, ∆U =U p+1−U p. (15)

For the first subiteration, p = 1, U p = Un and as p→ ∞, U p→Un+1. The spatial derivatives in the implicit (left-hand-side) op-erators are represented using standard second-order centered ap-proximations whereas high-order discretizations are employedfor the explicit terms (right-hand side). Although not shownin Eqn. 14, nonlinear artificial dissipation terms [22, 23] are ap-pended to the implicit operator to enhance stability. In addition,for improved efficiency, the approximately-factored scheme isrecast in diagonalized form [24]. Any degradation in solutionaccuracy caused by the second-order implicit operators, artificialdissipation and the diagonal form are eliminated through the useof subiterations. Typically, two to three subiterations are appliedper time step.

Implicit Large Eddy Simulation MethodologyThe high-order ILES method to be used in the present com-

putations was first proposed and investigated by Visbal et al [10].The underlying idea behind the approach is to capture with highaccuracy the resolved part of the turbulent scales while providingfor a smooth regularization procedure to dissipate energy at therepresented but poorly resolved high wave numbers of the mesh.In the present computational procedure the 6th-order compactdifference scheme provides the high accuracy while the low-passspatial filters provide the regularization of the unresolved scales.All this is accomplished with no additional sub-grid scale modelsas in traditional LES approaches. An attractive feature of this fil-tering ILES approach is that the governing equations and numer-ical procedure remain the same in all regions of the flow. In ad-dition, the ILES method requires approximately half the compu-tational resources of a standard dynamic Smagorinsky sub-gridscale LES model. This results in a scheme capable of capturingwith high-order accuracy the resolved part of the turbulent scalesin an extremely efficient and flexible manner.

Boundary ConditionsThe boundary conditions for the flow domain are prescribed

as follows. At the solid surface, the no slip condition is applied,requiring that the fluid velocity at the wing surface match the sur-face velocity. In addition, the adiabatic wall condition, ∂T

∂n = 0,and the normal pressure gradient condition ∂ p

∂n = 0 are specified.The treatment of the farfield boundaries is based on the ap-

proach proposed and evaluated previously in Reference 25 forsome acoustic benchmark problems. This method exploits theproperties of the high-order, low-pass filter in conjunction witha rapidly stretched mesh. As grid spacing increases away fromthe region of interest, energy not supported by the stretched meshis reflected in the form of high-frequency modes which are an-nihilated by the discriminating spatial filter operator. An effec-tive “buffer” zone is therefore created using a few grid points ineach coordinate direction to rapidly stretch to the farfield bound-ary. No further need for the explicit incorporation of complicatedboundary conditions or modifications to the governing equationsis then required. Freestream conditions are specified at the in-flow, side, upper and lower boundaries, while simple extrapola-tion of all variables is used at the outflow.

When solving the flapping wing problem, the mesh must beallowed to move in accordance with the motion of the wing sur-face. A simple algebraic method described in Ref. 26 deformsthe aerodynamic mesh to accommodate the changing wing sur-face position.

Aerodynamic Features of Flapping FlowAerodynamic computations have been performed for the

previously described rigid wing in isolation undergoing an anti-

5

FIGURE 8. Aerodynamic grid in the wing surface plane and the wingtip plane. Mesh beyond the wingtip and below the wing is not shown.

clockwise 8 cycle kinematics. The mesh system developed, Fig-ure 8, consists of 249 points in the streamwise direction, 289points in the spanwise direction and 151 points above and belowthe wing. 61 points in the chordwise direction and 141 pointsin the spanwise direction are located on the wing with a maxi-mum spacing of ∆y= 0.03 and ∆z= 0.028. A minimum spacing,∆x = 0.00025, is specified at the wing surface. The streamwisespacing downstream of the wing was restricted to a maximum of∆z = 0.04 for 1.85 chord lengths before the mesh was stretchedto the downstream boundary. This grid was subdivided into 200overlapping meshes for parallel processing.

The aerodynamic parameters specified for the flapping wingin forward flight consisted of a Reynolds number based on thefreestream velocity of Re = 1250. The advance ratio specified isJ = V∞/U f lap = 0.5 where U f lap = 2.0ΦR/T where R the dis-tance from the flap axis to the wing tip and T the period of theflapping motion. The Reynolds number based on the flappingvelocity is Re = 2500. The frequency of the flapping motioncorrespond to a Strouhal number, St = 0.2387 which gives a aperiod for the motion of 4.19 characteristic times. A time step,∆t = 0.00025 is employed based on the temporal resolution re-quired for the fine scale flow features observed.

Figure 9 presents the variation of the thrust coefficient, CTand the lift coefficient, CL over one period of the flapping mo-tion where time t∗ = 0 corresponds to the top of the upstroke andt∗ = 0.5 corresponds to the bottom of the downstroke. Distinctpeaks in the two curves are noted at the midpoints of the down-stroke and upstroke. An additional peak in the CL curve is seennear the bottom of the downstroke. The present flapping mo-tion produces thrust over nearly the full flapping cycle whereaspositive lift is produced during the upstroke and negative lift isproduced during the downstroke. The mean lift generated duringthe flap cycle is CL = 1.69 and the mean thrust is CT = 1.91. Thelabeled circles in Fig. 9 correspond to temporal locations whereflow visualizations are presented.

FIGURE 9. Temporal variation of the lift and thrust coefficient duringone flapping cycle

The flow structure during one flapping cycle is shown in Fig-ures 10 (downstroke) and 11 (upstroke). Figures 10a is at the topof the upstroke (pronation). On the upper surface of the winga small shallow separated vortical flow has formed outboard to-wards the tip of the wing. A very small tip vortex has also startedto form. On the underside of the wing the remnants of the vorti-cal structures previously created during the upstroke and the shedtrailing vortex are seen. At this point in the cycle only very lowlevels of force are produced on the wing. As the wing progressesthrough the downstroke the remnants of the previous vorticity onthe underside of the wing and the shed trailing edge vortex con-tinue to be convected downstream and out of the system. On theupper surface a strong leading edge vortex and tip vortex developas the downstroke commences. During this initial portion of thedownstroke, Figs. 10a-c, these vortices grow in strength and ex-tent but remain pinned at the wing corners. Extensive regionsof strong lift and thrust force are produced due to the low pres-sures that develop underneath these vortices on the upper surface.These forces produce the first peak in the lift and thrust seen inFig. 9.

As the downstroke continues several interesting new fea-tures emerge, Figure 10c-e. A trailing edge and wing root vor-tex are now clearly seen to have developed. Outboard on thewing the leading edge vortex breaks down and small scale vorti-cal structures are observed in the shear layer that rolls up to formthe leading edge vortex. These shear layer vortical structures canbe more clearly seen in Figure 12. The origin of these struc-tures is an unsteady interaction of the leading edge vortex withthe surface boundary layer. The proximity of the leading edgevortex to the wing surface and the resulting adverse pressure gra-dient leads to the separation of the surface boundary layer andthe formation of a secondary vortical structure (see Figure 12 in-set). This structure becomes unstable resulting in an unsteadyeruption of vorticity from the wing surface which interacts withthe shear layer separating from the leading edge. This unsteady

6

FIGURE 10. Flow structure (isosurface of q-criterion colored by pressure coefficient) and force distribution during downstroke portion of flappingcycle. Column 1 shows the flow on the upper surface, Column 2 shows the flow on the lower surface and Column 3 shows the distribution of the liftand thrust force at each time step. a)-e) correspond to the points a)-e) in Fig. 9

eruptive process drives the formation of the shear-layer substruc-tures. These shear-layer instabilities have been observed previ-ously in delta-wing flows [27, 28] and for revolving wings [5].A detailed description of the origin these structures is found inReference 28.

Figure 13 portrays the breakdown of the leading edge vortex

on a combination of two vertical planes through the core of thevortex to capture the curved structure. The intact core of theleading edge vortex is represented by the dark region of highentropy. A rapid decrease in entropy occurs at the point of vortexbreakdown. This point also corresponds to the start of a region ofreversed axial flow. These are typical characteristics of the onset

7

FIGURE 11. Flow structure (isosurface of q-criterion colored by pressure coefficient) and force distribution during upstroke portion of flapping cycle.Column 1 shows the flow on the upper surface, Column 2 shows the flow on the lower surface and Column 3 shows the distribution of the lift and thrustforce at each time step. f)-j) correspond to the points f)-j) in Fig. 9

of vortex breakdown.

Proceeding towards the bottom of the downstroke the lead-ing edge, wing tip and wing root vortices unpin from the twooutboard corners and the trailing edge corner at the root. Thevorticity is reoriented forming a ring-like structure that is pinnedat the leading edge corner at the wing root. This detached vor-

tex structure moves away from the wing surface and convectsdownstream with a corresponding reduction in both the lift andthrust forces. By the time the wing reaches the bottom of thedownstroke the outboard portion of the vortex has convected fur-ther downstream and the inboard sweep of the upstream portionis terminated when the trailing edge is reached with the vortex

8

FIGURE 12. Detailed representation of observed shear layer instabilities. Figure plots contours and isosurfaces of spanwise vorticity. Cycle timecorresponds to corresponds to Fig. 10d

being rapidly turned away from the wing surface, Fig. 11f. Ashallow separation region over the outboard portion of the wingand a wing tip vortex also reform at this point. The reformed vor-tical flow from the leading edge and a region of increased forcewhere the original leading edge vortex departs from the wing sur-face at the trailing edge results in a brief enhancement of the liftforces, Fig. 11f. This gives rise to the secondary peak in the liftforce around t∗ = 0.5 noted in Figure 9.

As the wing rotates and commences the upstroke portion ofthe motion the separated flow over the upper surface weakensand the remaining vorticity is shed and convected downstream,Figs. 11 g-h. On the underside of the wing a process similarto that observed on the upper surface for the initial portion of

the upstroke is observed with the formation of a leading edge,wake and tip vortex, which are initially pinned at the corners. Inthe wake of the wing a series of small scale vortices also form,Fig. 11h-i. As the upstroke progresses the tip vortex unpins fromthe trailing edge corner and connects with the trailing edge vor-tex. The leading edge vortex breaks down as the upstroke con-tinues, Fig. 11i-j, and also unpins from the leading edge cornerand reconnects with the tip vortex resulting in a complex sepa-rated flow region covering the the outer portion of the wing. Atthe top of the upstroke the leading edge separation weakens andultimately the vorticity is shed and convected downstream.

The vortex dynamics and resulting lift and thrust behaviorjust described can be understood in part by examining the vari-

9

FIGURE 13. Leading edge vortex breakdown visualized on a planethrough the vortex core using contours of entropy and reversed axialflow along the core of the vortex. Cycle time corresponds to correspondsto Fig. 10e

FIGURE 14. Temporal variation of the effective angle of attack forvarious locations along the leading edge

ation of the effective angle of attack and aerodynamic angle ofattack during the flap cycle. Figure 14 displays the effective an-gle of attack at various locations along the wing leading edgeduring one flap cycle. During the initial portion of the down-stroke a large effective angle of attack develops over most of the

FIGURE 15. Effective angle of attack at the wing tip, aerodynamicangle of attack, lift and thrust coefficients during one flap cycle

wing varying from αeffa ≈ 30 at 1/8th span to α

effa ≈ 85 at the

wing tip. This large effective angle of attack results in the de-velopment of the leading edge and tip vortices described earlier,Figures 10 a-c, and the resulting force produced. The reason bothlift and thrust occur can be further understood from Figure 15that compares α

effa and α

aeroa with the lift and thrust coefficient

for one flap cycle. Figure 15 shows that the thrust produced onthe downstroke results from the negative aerodynamic angle ofattack which tilts the force vector providing a component in thethrust direction. During the second half of the downstroke theeffective rapid pitch down motion particularly on the outboardportion of the wing, Figure 14, results in the shedding of the ini-tial vortex developed and the development of a new leading edgevortex, Figures 10d-e. This new leading edge vortex gave riseto the development of the secondary peak in the lift coefficient.Since over this portion of the cycle the aerodynamic angle of at-tack is small and mostly positive, Figure 15, minimal impact onthe thrust coefficient is noted.

During the upstroke the wing attains a positive aerodynamicangle of attack with a larger magnitude than the that attained dur-ing the downstroke. A negative effective angle of attack developswhich combined with the aerodynamic angle of attack again re-sults in the development of a strong vortex system on the under-side of the wing and wing thrust due to the force vector tilting. Itis interesting to note that at the wing root the effective angle ofattack is always positive during the upstroke. This accounts forthe lack of a wing root vortex on the underside of the wing dur-ing the upstroke. Also the underside leading edge vortex loses

10

its strength and coherence near the wing root. During the latterstages of the upstroke the positive effective angle progresses con-tinually outboard leading to the loss of the leading edge vortexon the underside of the wing for locations further outboard.

This description of the unsteady vortex dynamics generatedby this anti-clockwise 8 cycle is consistent with the features de-scribed by Viswanath and Tafti [14] for a similar kinematics al-beit for a higher Reynolds number. A number of the vortical fea-tures and dynamics noted by Garmann et al [5] for a revolvingwing have also been observed for the present case.

ConclusionsComputations for an aspect ratio 3.5 rectangular wing un-

dergoing an anti-clockwise figure 8 flapping motion and forwardflight has been presented. The high-order implicit LES schemeemployed for the computations exploits a sixth-order compactscheme to capture the resolved part of the turbulent scales. Veryhigh-order low-pass spatial filters provide the regularization ofthe unresolved small scales. A Reynolds number based on thefree stream velocity, Re = 1250 was specified with an advanceratio, J = 0.5.

A detailed description of the unsteady vortex developmentand structure during a complete flapping cycle was presented.The unsteady vortex dynamics generated both lift and thrust asa result of the flapping motion. Large effective angles of attack,which gave rise to strong vortical structures, and force vectortilting provided by the aerodynamic angle of attack allowed forthrust production during both the downstroke and upstroke of theflapping cycle.

ACKNOWLEDGMENTThis work was sponsored by the Air Force Office of Scien-

tific Research under a tasks monitored by Dr. Doug Smith. Thiswork was supported in part by a grant of HPC time from theDoD HPC Shared Resource Centers at the Air Force ResearchLaboratory.

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for Electromagnetics”. AIAA Paper 97-0363 Jan. .[20] Alpert, P., 1981. “Implicit Filtering in Conjunction with

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[22] Jameson, A., Schmidt, W., and Turkel, E., 1981. “Nu-merical Solutions of the Euler Equations by a Finite Vol-ume Method Using Runge-Kutta Time Stepping Schemes”.AIAA Paper 81-1259 .

[23] Pulliam, T., 1986. “Artificial Dissipation Models for theEuler Equations”. AIAA, 24 (12) , pp. 1931–1940.

[24] Pulliam, T., and Chaussee, D., 1981. “A Diagonal Form ofan Implicit Approximate-Factorization Algorithm”. Jour-nal of Computational Physics, 39 (2) , pp. 347–363.

[25] Visbal, M., and Gaitonde, D., 2001. “Very High-OrderSpatially Implicit Schemes for Computational Acoustics onCurvilinear Meshes”. Journal of Computational Acoustics.

[26] Melville, R. B., 2000. Dynamic Aeroelastic Simulationof Complex Configurations Using Overset Grid Systems.AIAA 2000-2341, June.

[27] Gordnier, R. E., and Visbal, M. R., 1994. “Unsteady VortexStructure over a Delta Wing”. Journal of Aircraft, 31 (1)Jan.-Feb. , pp. 243–248.

[28] Visbal, M. R., and Gordnier, R. E., 1995. “Origin of Com-puted Unsteadiness in the Shear Layer of Delta Wings”.Journal of Aircraft, 32 (5) , pp. 1146–1148.

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