Flow simulations with ultra-low Reynolds numbers over rigidand flexible airfoils subject to heaving and flapping motions

D. Antonelli1,3†, C. Sacco2, and J. Tamagno1

1 Cordoba National University, Cordoba, Cordoba, 5000, Argentina2 Aeronautical University Institute, Cordoba, Cordoba, 5000, Argentina

3 Consejo Nacional de Investigaciones Cientificas y Tecnicas-CONICET, Argentina

† Corresponding Author Email: dinoantonelli@hotmail.com

(Received ——; accepted ——)

ABSTRACT

Numerical simulations of flow patterns at ultra-low Reynolds numbers over rigid and flexible airfoils andthe influence of flexibility on main aerodynamic properties, are presented and discussed. Typical unsteadyflights like heaving and flapping are, in terms of Reynolds and Strouhal numbers, reduced frequencies andFSI (Fluid Structure Interaction) factor, are valuated. It has been found that for some flexibility levels, theaerodynamic forces and propulsive efficiency are enhanced if compared with a rigid airfoil. The mathe-matical technical approach used to solve the laminar-incompressible flow equations coupled with structuralalgorithms, is described.

Keywords: Aerodynamic wing sections, CFD, Fluid structure interaction, Unsteady flows, Low Reynolds,Flexible airfoil, Partitioned method, Finite element method.

NOMENCLATURE

c chordc convective velocityCD drag coefficientCL lift coefficientCP power coefficientCT thrust coefficientcx generic average coefficiente structural thicknessE Young modulusfh heaving frequencyfα pitching frequencyha heaving amplitudeI Inertial momentk reducy frequency, iterationL liftM Mach numberP input powerp fluid pressureRe Reynolds numberU reference velocitySt Strouhal numbert timeT Thrustu fluid velocity

us interface structural velocityu f interface fluid velocityum mesh velocityu fractionary velocityw transversal structural displacementw interface displacementα angle of attackδ∗ flexibilityδ tip displacement∆t time stepη propulsive efficiencyµs structural mass per unit lengthρ f fluid densityρs structural densityρ∗ density relationΣ FSI intensity factorσ fluid stress tensorφh test elemental functionχα pitching phaseχh heaving phaseψh test elemental functionω relaxation Aitkens factorΩ analysis domain

1. INTRODUCTION

The importance of ultra-low Reynolds flows lies intechnological applications like MAVs (Micro AirVehicles). They are flying systems with maximumdimensions of 0.15[m] that can lead to revolutio-

nary improvements in remote sensing and informa-tion gathering capabilities both in military as wellin civilian applications Radmanesh et al. (2014).In many cases, an in depth study of phenomenaobserved in flight is necessary to obtain maximumpropulsion with the highest efficiency.

Because of Reynolds number effects, aerodynamiccharacteristics such as lift, drag and thrust of aflight vehicle change considerably between MAVsand conventional manned air vehicles. In fact, inthe nature, birds or insects flap their wings interac-ting with the surrounding air to generate lift to stayaloft or producing thrust to fly forward. The mainpowered flights are: heaving and flapping (flightswith free stream) and hovering (flight without freestream).

Unsteady aerodynamic mechanisms such as thegeneration of a leading edge vortex (LEV), wing-wake interaction, and three-dimensional flow fea-tures, such as tip vortex-vortex interactions, all sig-nificantly affect the aerodynamic force generation.Another remarkable mechanism that the biologicalflyers seem to be using is the wing flexibility. Stu-dies have been performed to shed light on the in-terplay between the structural flexibility and the re-sulting aerodynamic forces.

Much research in this broad area have been made.The most significant that can be named are: Guer-rero (2008) carried out unsteady aerodynamic stu-dies at ultra-low Reynolds in 2D and 3D configu-rations built using the NACA 0012 wing section;Combes and Daniel (2005) have shown that a varie-ty of insects exhibit anisotropy in their wing struc-tures based on static response tests.

Experimental and numerical studies Kang and Shyy(2012), Kang et al. (2011), Aono et al. (2009)have shown that the chordwise flexibility affects thedistribution of the resulting aerodynamic forces inlift and thrust directions. For example, if the plateshape undergoes deformation then the camber ofthe plate may change, leading to an effective geo-metry modification, wich combined with the pit-ching angle the direction of the net force can beadjusted in favor of the thrust generation. Further-more, for a range of spanwise flexibility, deformedairfoil shapes along the spanwise and the correct di-rection of the motion, are seen to enhance the thrustof a plunging wing.

Zhu (2007) showed numerically that the thrust andthe propulsive efficiencies increased more for aplunging chordwise flexible airfoil in water thanimmersed in air. Hence it is seen that the flexibility,including the density ratio can be utilized to controlresulting aerodynamic forces. However the preciseunderlying physics of aeroelastic coupling for flap-ping wings and its applicability to the MAV designsare yet to be understood.

The aerodynamic force generation caused by struc-tural flexibility, is definitely essential to change lo-cal behaviors in thrust and propulsive efficiency.Studies like Olivier (2010), Heathcote and Gursul

(2005), Chandar and Damodaran (2009), Naderiet al. (2016) analized flexibles and rigid airfoilsunder sinusoidal flapping motion (or combined pit-ching and heaving) of flexible insect wings un-der realistic flight conditions such as forward flightand/or rapid maneuvering.

Ranges of non-dimensional numbers found rele-vant to unsteady flights of biological ”flappers”, arealso considered valid for MAVs. A characteristicone for flapping motions is the Strouhal numberSt = 2 fhha/U , where fh is the frequency and ha theamplitude of motion. Therefore, the Strouhal num-ber expresses the ratio between the flapping wingvelocity and the reference velocity U . The reducedfrequency given by k = π fhc/U is another parame-ter that can be interpreted as a measure of unsteadi-ness comparing the wave length of the flow distur-bance to the chord c. Then the fundamental aero-dynamics parameters are computed by:

CL(t) =L(t)

12 ρ fU2c

(1)

where CL is the lift coefficient. Then the thrust co-efficient is:

CT (t) =−CD(t) =T (t)

12 ρ fU2c

(2)

where T =−D. and the input power coefficient is:

CP(t) =P(t)

12 ρ fU3c

(3)

The thrust efficiency is:

η =TUP

=ctcp

(4)

where ct and cp are the average thrust and powercoefficents respectively.1

The dimensionless parameters of the flexible flap-ping wing sections problem are Olivier (2010):flexibility,

δ∗ =

ρ f ( fhha)2c3

EI(5)

wich relates the dynamic pressure with structuralstiffness and the FSI intensity factor is defined by,

Σ =ρ f ha

ρse(6)

1The average coefficents are computed by: cx =1T∫ t+T

t CX (t)dt where T is the period of the induced oscillatorymotion and CX (t) can be P(t), T (t), CL(t), CT (t), etc.

2

wich represent the relation between structural iner-tial forces and aerodynamic pressure. Density rela-tion is:

ρ∗ =

ρs

ρ f(7)

2D unsteady flow sinusoidal kinematics are givenby the equations:

h(t) = hasin(2π fht +χh) (8)

α(t) = αasin(2π fαt +χα) (9)

where χh and χα are the phases angles.

2. NUMERICAL METHOD

2.1 Fluid Module

The two-dimensional time-dependent Navier-Stokes equations are solved using the finiteelement method, assuming incompressible-laminarflow which is justified since the Mach numberof a MAV flight is M << 0.3 and the Reynoldsnumber Re < 10000. To represent the unsteadyflow, the Navier-Stokes equations are solved ina fixed inertial reference frame incorporatinga moving mesh with velocity um following theArbitrary Lagrangian Eulerian (ALE) formulationDonea and Huerta (2003). Conservation of massand momentum in a Ω analysis domain withboundaries Γu

⋃Γσ, with (t0, t f ) time interval of

analysis, are described by:

∇.u = 0 (10)

∂(u)∂t

+(c.∇)u+1

ρ f∇p−ν∇

2u− fe = 0 (11)

The u is the two-dimensional flow velocity vector,ρ f the constant density, ν the kinematic viscosity, pthe pressure, fe the external forces and c = u−umis the convective velocity that represent the diffe-rence between fluid velocity and mesh velocity. Inthe present work, the algorithm of mesh movementis based in operations of optimal smoothing, deve-loped by Canann et al. (1998).

The equations previously presented can not besolved by a numerical standard form because in-compressibility gives raise to a flow field restric-tion. There are several algorithms to deal with thisdifficulty and the Fractional Step method used hereis one of them. The method meets the LBB con-dition Babuska (1971) through the use of same or-der of approximation for velocity and pressure. Toapply the Fractional Step algorithm the momentum

equation 11 without mesh movement um = 0 is di-vided in two parts,

un+1 = un+

δt[

un+θ.∇un+θ + γ1

ρ f∇pn−ν∇

nun+θ + fn+θ

](12)

un+1 = un+1− δtρ f

(∇pn+1− γ∇pn) (13)

In the Eq. 12 the fractionary velocity u is intro-duced and used in Eq. 13. If the divergence of Eq.13 is taken and the continuity equation is applied,results:

∇2(pn+1− γpn) =

ρ f

δt∇.un+1 (14)

Through this equation the pressure is calculated. Inaddition, γ is a numerical parameter such that itsvalues of interest are 0 and 1. The θ parameter de-termine the kind of temporal approximation.

Then the Finite Element Method is used to dis-cretize the govern equations and it provides an ap-propriate resolution procedure Lohner (2001). Theresultant scheme is of first order (γ = 0) and thetemporal discretization (θ = 0) results in Euler for-ward. The test functions (ψh,φh) ∈ Ψh ×Φh areused such as:

1δt(un+1

h ,ψh) =1δt(un

h,ψh)− (unh.∇un

h,ψh)−

ν(∇unh,∇ψh)− (fn

e ,ψh) (15)

(∇pn+1h ,∇φh) =

ρ f

δt

[(un+1

h −unh,∇φh)− (∇un

h,φh)]

(16)

(un+1h ,ψh) = (un

h,ψh)−δtρ f

(∇pn+1h ,φh) (17)

where subindex h means it is applied on one ele-ment. The last equations system is semi-implicitbecause Eqs. 15 and 17 are explicit (lumped massmatrix) and Eq. 16 for the pressure computation isimplicit 2.

The discretization of convective terms yields nu-merical instabilities, therefore stabilization me-thods must be used. In this work the Orthogo-nal Subgrid Scale (OSS) algorithm is applied Co-dina (2000b), Codina (2000a), Principe and Codina

2The notation used in the equations means: (a,b) =∫

a.bdΩ

3

(2009). The expresion for the convective stabiliza-tion term ST Bu is:

ST Bu = τ1(unh.∇un

h−πnh,u

nh∇ψh) (18)

where πnh is the convective term projection and it is

defined in Eq. 23. This equation add to momentumEq. 15 and it is evaluated in tn, therefore it remainsexplicit.

The term stabilization of pressure ST Bp to be addedto the Eq. 16 is:

ST Bp =−(τ2(∇pn−ξnh),∇φh) (19)

where ξnh is the gradient pressure term projection

and it is defined in Eq. 24. In addition, it is evalua-ted in tn, therefore it remains explicit.

The final complete stabilized scheme with meshmovement um is obtained:

1δt(un+1

h ,ψh) =1δt(un

h,ψh)− (cnh ·∇un

h,ψh)

−ν(∇unh,∇ψh)− (fn

e ,ψh)

− (τ1(cnh ·∇un

h−πnh),∇cn

h ·∇ψh) (20)

(∇pn+1h ,∇φh) =

ρ f

δt + τ2

[(un+1

h −unh,∇φh)− (∇un

h,φh)]

+τ2

δt + τ2(∇ξ

nh,∇φh) (21)

(un+1h ,ψh) = (un

h,ψh)−δtρ f

(∇pn+1h ,φh) (22)

(πnh, ψh) = (cn

h.∇unh, ψh) (23)

(ξnh, ψh) = (∇pn

h, ψh) (24)

where ψh ∈ Ψh and τ1, τ2 are stabilization coeffi-cents. The system of equations of Eqs. 20, 22, 23,24 are solved in explicit form with lumped massmatrix and the system resultant of Eq. 21 is solvein explicit form through of conjugate gradients withdiagonal pre-conditioner.

It is noted that the formulation of the scheme isn’t inthe ALE framework. To account the mesh velocityit is necessary to introduce the convective velocityc in convective and stabilization terms.

Finally the boundary conditions in viscous tensorand velocity are:

• Imposed velocity: u = uc

• No slip: u = 0

• No traction: n.σ.n = 0

where σ is the fluid stress tensor and n a normalsurface versor.

2.2 Structural Module

The govern equation of the structural model is:

∂2

∂x2

(EI

∂w2

∂x2

)=−µs

∂w2

∂t2 +q(x) (25)

where w is the transverse displacement, µs is themass per unit length, E Young’s modulus, I inertialmoment and q(x) a distributed transverse load.

Computations done for a flexible airfoil composedof a rigid teardrop and elastic flat plate at higherReynolds number and for various motion frequen-cies, Kang et al. (2011) showed that a linear Euler-Bernoulli beam is sufficient for the analysis of thefluid-structure interaction.

The Euler-Bernoulli beam model has been incorpo-rated to solve Eq. 25 using a finite element (FE)representation. The structural damping is not con-sidered in this study and two degree of freedom,i.e. displacement and bending, are allowed at eachnode. The FE solution approach Cook et al. (2001),Wright and Cooper (2007) is given by the followingsteps:

• Determine the dynamic properties of each ele-ment in the form of element stiffness and massmatrices. In order to write the strain energyand the kinetic energy terms for the element,the variation of displacement within the ele-ment will need to be expressed as a function ofthe nodal displacements. It is assumed that thevariation of the transverse displacement alongthe beam elements can be expressed as a cubicpolynomial.

• Assemble all the elements to form global massand stiffness matrices from which modes andresponses may be determined. The assem-bly process satisfies exact compatibility of dis-placements/rotations between elements. Theadvantages of the finite element method arethat more elements may be used in regionswhere the displacement and/or stress is ex-pected to vary more rapidly and that morecomplex geometries and problems may behandled. The general equation to solve is:

Mr r+Krr = R (26)

where Mr is the global mass matrix, r is thedisplacements vector, Kr is the global stiffnessmatrix and R is the external forces vector.

4

• Solve the general eq. 26 via damped Newmarktime integration scheme.

2.3 Coupling Strategy

The fluid-structure interaction is based on a time-domain partitioned solution process, in which thepartial differential equations governing the fluidand the structure are solved independently and spa-tially coupled, through the interface between thefluid and the structure Degroote (2010). At eachtime step the fluid F and structural S solvers arecalled one after the other, until sufficient conver-gence on the displacements on the shared boundarysurface are reached in an inner-iteration before ad-vancing to the next time step.

Since both fluid and structure have been mode-led with continuous theory, the treatment of thefluid-solid interface makes no exception. Only themass conservation and the momentum conservationequations are considered as the other physics prin-ciples are not required to describe elastic solid andnearly incompressible flows. Therefore, the mo-mentum and mass conservation at the FSI interfaceyields the following conditions Olivier (2010):

us = u fσ f ·n = σs ·n

(27)

which means that the fluid ( f subindex) and struc-tural (s subindex) velocities and normal loads, areequals in the interface. These conditions are imple-mented in the code following the method presentedin Maza et al. (2012).

In this work, the strong coupling block Gauss-Seidel partitioned method with relaxation Vazquez(2007), Kuttler and Wall (2008) has been imple-mented. Starting from known values of fluid struc-ture and mesh in time tn, a scheme of the couplingalgorithm is presented in Fig. 1.

3. VERIFICATION OF NUMERICAL CODE

3.1 Verification of Baseline Solvers

As a verification case of the fluid solver, a rigidNACA 0012 airfoil in flapping motion is analyzed.The following parameters are considered: pitchingand heaving frequencies fα = fh = 0.225Hz, re-ducy frequency k = 0.7096, maximum heaving am-plitude ha = 1, phase angle χα = 90, Strouhalnumber St = 0.45 and the variable parameter is thepitching amplitude αa.

In the Table 1 a comparison of maximum lift coef-ficient CLm and average thrust coefficient ct is pre-sented. It can be concluded that the results obtainedin this work compare well with those given by Pe-dro et al. (2003) and Guerrero (2008).

Advance time step: tn+1 =tn + ∆t. Set iteration k = 1.

Structural predictor Neumann-Dirichlet (Order 2): Find theinterface displacement wn+1

kfrom structural solver S with

a predicted external forcegiven by a fluid pressure load:pn+1 = 2pn− pn−1 to n ≥ 2.

Iterate the coupled FSI problem

Mesh movement: Transfer wn+1k

to the mesh solver. Find theaerodynamic nodal displacements

and mesh velocities umn+1k .

Solve the fluid-dynamic equa-tions (F): Transfer the mesh

velocities umn+1k to fluid solver

F. Solve the fluid-dynamicproblem and find the fluid ve-

locities un+1k and pressures pn+1

k .

Solve the structural equations(S): Transfer the aerodynamic σ

and inertial loads to structural code.Solve the structural problem and

find the beam displacements wn+1k+1 .

Relaxation fase: Compute optimalrelaxation parameter ωk via

Aitken’s method. Relaxation ofpredicted interface position with,wn+1

k+1 = (1−ωk)wn+1k +ωkwn+1

k+1 .

Incrementiterationk = k+ 1

Check con-vergence.

Go to nexttime step

no

yes

Fig. 1. Chart of fluid-structure interaction cou-pling algorithm.

5

Table 1 Comparison of average thrustcoefficient ct and maximum lift coefficient CLm

in flapping motion for rigid NACA 0012.Pedro et al. Guerrero P. Work

αa ct CLm ct CLm ct CLm5 0.43 8.33 0.42 8.08 0.43 8.2110 0.65 7.48 0.66 7.17 0.66 7.2415 0.82 6.63 0.84 6.54 0.82 6.3920 0.93 5.82 0.94 6.11 1.00 5.5125 1.00 5.06 0.96 5.61 1.09 4.99

Fig. 2. Comparison of tip displacement of beamwith prefixed displacement.

As a case for validating the solid module, a vibra-ting cantilever beam model with prefixed displace-ment is analyzed. This problem were solved byHan et al. (1999) with four differents theoreticalmodels (Euler-Bernoulli, Shear, Rayleigh and Tim-oshenko), through the method of eigenfunction ex-pansion.

The beam properties are: length L = 1m, tubu-lar section: internal ratius ri = 0.15m, externalratius re = 0.16m, section area A = 0.0097389m2,area inertial moment I = 0.0001171m4 and densityρs = 7830kg/m3. Twenty finite unidimensional e-lements along the beam were used. The function ofinitial transversal displacement w(x,0) is:

w(x,0) = (1.667x3−5x2)10−3 (28)

where x is the coordinate along beam.

The tip displacements of the beam due to differentsmethods are presented in the Fig. 2. The agreementwith the analytic results can be observed.

3.2 Verification of FSI Solver

The present FSI solver is verified with a slenderflexible structure fixed at the downstream end of abluff body. The body generates vorticity wich in-duces oscillations in flexible structure. This prob-lem was proposed originally by Wall and Ramm(1998). The domain and boundary conditions are

Fig. 3. Problem FSI domain specifications (out ofscale).

Fig. 4. Comparison of tip displacement of flexiblestructure fixed at bluff body.

presented in Fig. 3. The material properties ofthe structure are density ρs = 0.1Kg/m3 and Youngmodulus E = 2.5e6. The flow properties are den-sity ρ f = 1.18e− 3Kg/m3, dynamic viscosity µ =1.82e − 4Kg/m.s and free stream velocity U =51.3m/s. The Reynolds number is Re = 333 withthe side length of the square rigid body used likecaracteristic length Lb = 1m. The fluid-structureinteraction is strong with a the density relation ofρ∗ = 84.74.

Table 2 Comparison for FSI problemP. Work Kassiotis Valdez Wall

f f s[1/s] 3.186 3.175 3.125 3.071err f [%] - 0.352 1.940 3.700

δ[m] 0.998 1.031 0.997 1.328erra[%] - 3.200 0.10 24.80

The fluid-dynamic mesh domain has 24745 ele-ments and the structural mesh has 40 unidimen-sional elements. The Fig. 4, shows the periodicstates obtained by different authors Wall and Ramm(1998), Kassiotis et al. (2011), Vazquez (2007) incomparison with the present work. In this work, theperiodic state is assumed when the difference be-tween maximums amplitudes is lower than 0.05. Aclose agreements are observed between the present

6

Fig. 5. Average thrust coefficient ct and propulsive efficiency η for rigid airfoils in heaving motion (NACA0004, 0006 and 0012).

Fig. 6. Velocity contours for rigid NACA 0004 (a,b,c,d) and NACA 0012 (e,f,g,h) in heaving motion(St = 0.3).

Fig. 7. Average thrust coefficient ct and propulsive efficiency η as function of αa in flapping motion ofrigid airfoils (NACA 0004 and 0012).

7

computed results and the numerical values of theauthors, mentioned above.

To complete the analysis in Table 2 a comparisonbetween response frequencies f f s and maximum tipdisplacements δ with respective errors are listed 3.

4. NUMERICAL RESULTS

The fundamental parameter of unsteady analysis isthe Strouhal number, defined as St = 2 f ha/U . Tay-lor et al. (2003) and Triantafyllou et al. (1993)performed a study of wing frequencies and ampli-tudes, and cruise speeds across a range of birds, in-sects, fishes and cetaceans, to determine Strouhalnumbers in “cruising” flight. They found 75% ofthe 42 species considered fall within a narrow rangeof 0.19 < St < 0.41 Guerrero (2008). Therefore, asimilar range of Strouhal numbers in this work hasbeen selected.

4.1 Heaving motion of rigid airfoils

The first analysis is a heaving motion with kine-matic described by Eq. 8. The analysis is appliedto NACAs four digits (0004, 0006 and 0012) sym-metrical rigid airfoils. The kinematics parametersare: heaving frequency fh = 1Hz and a variableStrouhal number throughout the heaving amplitude0.1 < ha < 0.5 and Re = 1100. The average thrustcoefficient ct and propulsive efficiency η = ct/cpin terms of the Strouhal number are presented inFig. 5. Note that the numerical results obtained byGuerrero (2008) are added. It can be seen in Fig.5 that thrust and propulsive efficiency increase withairfoil thickness.

Comparisons between velocity contours of NACA0004 and NACA 0012 airfoils at differents timesFig. 6, helps to understeand the simulation resultsshown in Fig. 5. Times in Figs. 6 (a) and (e) ist = 0.45s, (b) and (f) t = 0.86s, (c) and (g) t = 1.29sand (d) and (h) t = 1.64s. The formation of leadingedge vortices (LEV) and its convection toward thewake, can be detected. With NACA 0004 the LEVremains longer time at the nose that it does withNACA 0012, and the low pressure created there de-lays convection of th vortex toward the wake andtherefore affecting the propulsive and its efficiency.

4.2 Flapping motion of rigid airfoils

The second analysis is the flapping motion. Thename flapping is applicable to a combined motionof heaving and pitching, consequently the kinema-tics relations given by sinusoidal Eqs. 8 and 9 aresimultaneously applied.

In Fig. 7 are shown average thrust coefficients ct

3The error was calculated like err f =fp− fr

frwhere fp is the

present work frequency and fr the referenced frequency. To themax. tip displacement errors the same form is applied.

and propulsive efficiencies η applicable to NACAsymmetric rigid airfoils 0004 and 0012, as func-tion of the pitching angle 5 < αa < 25 with fα =fh = 0.3, ha = 0.5m, χα = π/2 and St = 0.3. Thethrust and efficiency are positives in all cases ob-tained and greater than obtained in heaving motionfor rigid airfoils. Both geometries show a similarct curve although NACA 0012 is somewhat moreefficient than NACA 0004. In Fig. 8 are shownaverage thrust coefficients ct and propulsive effi-ciencies η applicable to NACA symmetric rigid air-foils 0004 and 0012, as function of the heavingamplitude 0.025 < ha < 0.5 (0.05 < St < 1) withfh = fα = 1, αa = 15 and χα = π/2. The ct coeffi-cient always increases with the heaving amplitude.The efficiency shows a tendency to increase up toha = 0.1 (St = 0.3) and then decreases for largeramplitudes.

4.3 Heaving motion of flexible airfoils

An analysis of the influence of flexibility δ∗ (Eq.5) and fluid-structure interaction intensity factor Σ

in two symmetric airfoils NACA 0012 and NACA0004 in heaving motion, is performed. The samesinusoidal kinematic and parameters of rigid air-foils are used. The Strouhal number is fixed toSt = 0.3 (maximum efficiency obtained in rigidairfoils). The variable parameters through Youngmodulus E are 8.707e− 6 < δ∗ < 8,707e− 3 andthrough structural density ρs, 0.217 < Σ < 1.783.

The average traction coefficient ct and the efficien-cy η are presented in Figs. 9 (NACA 0012) and10 (NACA 0004). It can be seen the same be-havior in both airfoils, although little higher va-lues for NACA 0004. When the structural flexi-bility is lower than δ∗ > 1e− 3, the coefficientsare almost equals to the rigid airfoils (the defor-mation is very small). When the flexibility δ∗ >1e− 3, ct increases to a maximum value and thendecreases. When the FSI intensity factor Σ is di-minished (structural density ρs is increased), ct isincreased. Note that efficiency η is dependent ofcp, that it presents a high decrease with δ∗.

As in the analysis of rigid airfoils, a comparison atdifferent times of velocity contours between flexi-bles NACA 0004 and NACA 0012 (Fig. 11) helpsto understand the results of the simulations. Thecase selected is δ∗ = 2,64e−003 and Σ = 0,26168.Times in Fig. 11 are the same used with rigid air-foils. These are (a) and (e) is t = 0.45s; (b) and(f) t = 0.86s; (c) and (g) t = 1.29s and (d) and (h)t = 1.64s. In both airfoils the leading edge vortexesare convected faster than rigid airfoils. A low pres-sure region is generated in the lower surface du-ring the upstroke and in the upper surface duringthe downstroke. In addition, the flexion of airfoilincreases the low presion and improves the convec-

8

Fig. 8. Average thrust coefficient ct and propulsive efficiency η as function of ha in flapping motion ofrigid airfoils (NACA 0004 and 0012).

Fig. 9. Thrust coefficient ct and propulsive efficiency η in heaving motion for flexible NACA 0012.

Fig. 10. Thrust coefficient ct and propulsive efficiency η in heaving motion for flexible NACA 0004.

9

Fig. 11. Velocity contours for flexible NACA 0004 (a,b,c,d) and NACA 0012 (e,f,g,h) in heaving.

Fig. 12. Thrust coefficient ct and propulsive efficiency η in flapping motion for flexible NACA 0012.

Fig. 13. Thrust coefficient ct and propulsive efficiency η in flapping motion for flexible NACA 0004.

10

tion of LEV toward the trailing edge and its cou-pling with the wake. The consequence is a increasein ct and η like the results are showing.

4.4 Flapping motion of flexible airfoils

4.41 Sinusoidal kinematic

The next analysis applies to flapping motion onflexible airfoils. The kinematics are sinusoidal andthe parameters of motion are St = 0.3 (ha = 0.15),Re = 1100, fh = fα = 1, αa = 10 and χα = π/2. Itis observed in Figs. 12 and 13 that in both airfoils,ct has increased with respect the value obtained inheaving motion. Maximum ct from flapping arelimited to the interval of flexibility 1e− 3 < δ∗ <1e− 2, and this interval is the same for maximumct produced in heaving motion.

Note that greater ct values are for minimum Σ orhigher ρs values. If the efficiency η is comparedwith the results obtained in flexible airfoils in hea-ving motion, the difference found between maxi-mums values of ct are not big, it can be concludedthat in certain cases the structural flexibility can re-place rotations mechanism in MAVs, saving someweight.

4.42 Alternative kinematic

A third analysis of flexible airfoils in flapping mo-tion with an alternative kinematic is performed.This kinematics is characterized by Eqs. 29 and 30:

h(t) =ha

sin−1(0.8)sin−1 (0.8 sin(2π fht)) (29)

α(t) =αa

tanh−1(3)tanh−1 (3 sin(2π fαt +χα)) (30)

The model takes the wing kinematic used in theRobofly model of Dickinson et al. (2004) and Boset al. (2007). Based on observation of true insectflights, it was accepted that the wing maintains aconstant velocity and angle of attack during most ofthe stroke, with a relatively strong linear and angu-lar acceleration during stroke reversal. This resultswith typical “sawtooth” displacement and “trape-zoidal” angle of attack pattern of the Robofly kine-matic model, are illustrated in Fig. 14. The pa-rameters of kinematic are the same used in sinu-soidal kinematic, St = 0.3 (ha = 0.15), Re = 1100,fh = fα = 1, αa = 10 and χα = π/2.

In the Figs. 15 and 16 the average thrust ct and ef-ficiency η are presented. In both airfoils is detectedan increase in the thrust respect to the sinusoidalkinematic, and the maximum values are between1e− 4 < δ∗ < 1e− 3. The greater ct values (mini-mum Σ) are given by the NACA 0004 airfoil. In the

Fig. 14. Alternative kinematic.

alternative kinematic, the angle of attack remains inits position of maximum amplitude for longer time,because of its “trapezoidal” pattern (Fig. 14). Thisimplies that vortices generated at the trailing edge(TEV) are more intense, as a results high speedthere is in the wake. Therefore, in the leading edgeexist a high suction during more time in compari-son with sinusoidal kinematic wich allows betterconvection of generated (LEV). Since both edgesin NACA 0004 are sharper than in NACA 0012, theconvection is even more intense.

The propulsive efficiency presents a different be-havior respect to the sinusoidal kinematics and themaximums values are between 1e−4< δ∗ < 5e−3, however it depends of average power coefficientcp.

5. CONCLUSIONS

In heaving motion studies, rigid and flexible sym-metrical wing sections are considered. Averagethrust coefficients and propulsive efficiencies forselected motion frequencies are numerically simu-lated, and the results plotted in terms of a Strouhalnumber determined using the heaving amplitude.As a help to better understand the simulation re-sults, velocity contour pictures allowing compari-son between wing sections at prescribed identicaltimes, are built and shown. Based on these pic-tures, generation and displacement of vortexes asthe wing section executes the heaving motion aredescribed, and justifications about why an airfoilhas better performances than other one when exe-cuting such motion, are given. Studies related tothe influence of flexibility and FSI intensity fac-tor on thrust coefficients and propulsive efficienciesin heaving motions, are made. From comparisonswith rigid data it is concluded that for some flexi-bility values, improvements are feasible.

Combinations of pitching and heaving motions(flapping) for rigid and flexible wing sections, havealso been simulated. Like in rigid airfoils, the ma-ximum pitch angle and vertical displacement am-

11

Fig. 15. Average ct, cp and η coefficients in alternative flapping kinematic (NACA 0004).

Fig. 16. Average ct, cp and η coefficients in alternative flapping kinematic (NACA 0012).

12

plitudes are taken as plotting variables. It is foundthat for given pitch angles a maximum propulsionvalue exist, and this maximum increases when theheaving amplitude of the flapping motion, is alsoincreased. With flexible airfoils, a comparison be-tween two flapping kinematics is performed. Inboth kinematics, the influence of flexibility andFSI intensity factor on propulsion and its efficiencyare studied. Between the sinusoidal kinematic andheaving motion, a similar behavior is found. There-fore, the possibility exist of somehow using wingflexibility (passive pitching) to replace pitching mo-tion hardware in MAV applications and so, theweight can be reduced.

The Finite Element software here used can in runtime, build moving grids needed to perform nume-rical simulations of unsteady motions like heavingand flapping with rigid as well as with flexible 2Dwing sections. For dealing with flexible wing sec-tions, an appropriate structural module and algo-rithms to compute fluid-structure interactions (FSI),are implemented. In addition, and to handle com-plex geometries, more elements are created in re-gions where it is expected that displacements and/orstrains vary rapidly.

ACKNOWLEDGMENTS

To CONICET and Aeronautical departament ofUNC for the financial and facilities support to thisresearch.

REFERENCES

Aono, H., S. Chimakurthi, C. Cesnik, H. Liu,and W. Shyy (2009). Computational mod-eling of spanwise effects on flapping wingaerodynamics. AIAA 1270.

Babuska, I. (1971). Error bounds for finite ele-ment method. Numer. Math. 16, 322–333.

Bos, F., D. Lentink, B. van Oudheusden, andH. Bijl (2007). Numerical study of kinematicwing models of hovering insect flight. 45thAIAA Aerospace Sciences Meeting and Ex-hibit 482.

Canann, S. A., J. R. Tristano, and M. L. Staten(1998). An approach to combined laplacianand optimization-based smoothing for tri-angular, quadrilateral, and quad-dominantmeshes. International Meshing Roundtable.

Chandar, D. D. J. and M. Damodaran (2009).Computational fluid-structure interaction ofa flapping wing in free flight using overlap-ping grids. AIAA 2009-3676.

Codina, R. (2000a). Pressure stability in frac-tional step element methods for incompress-ible flows. Computational Physics 170.

Codina, R. (2000b). Stabilization of incom-pressibility and convection trough orthog-onal sub-scales in finite element methods.Computational Physics 190, 1579–1599.

Combes, S. A. and T. L. Daniel (2005). Flexu-ral stiffness in insect wings. effect of winsvenation and stiffness distribution on passivebending. American entomologist.

Cook, R., D. S. Malkus, R. J. Witt, and M. E.Plesha (2001). Concepts and Applications ofFinite Element Analysis. Hardcover.

Degroote, J. (2010). Development of Algorithmsfor the Partitioned Simulation of StronglyCoupled Fluid-Structure Interaction Prob-lems. Ph. D. thesis, Ghent University, Ghent,Belgium.

Dickinson, M. H., F. Lehmann, and S. P. Sane(2004). Wing rotation and the aerodynamicbasis of insect flight. Science 284, 195460.

Donea, J. and A. Huerta (2003). Finite ElementMethods for Flow Probelms. John Wiley andSons.

Guerrero, J. (2008). Numerical simulation of theunsteady aerodynamics of flapping flight. Ph.D. thesis, University of Genova, Genova,Italy.

Han, S. M., H. Benaroya, and T. Wei (1999). Dy-namics of transversely vibrating beams usingfour engineering theories. Journal of Soundand vibration.

Heathcote, S. and I. Gursul (2005). Flexible flap-ping airfoil propulsion at low reynolds num-bers. AIAA Paper 2005-1405-CP.

Kang, C., H. Aono, C. E. S. Cesnik, and W. Shyy(2011). Effects of flexibility on the aerody-namic performance of flapping wings. 6thAIAA Theoretical Fluid Mechanics Confer-ence.

Kang, C. and W. Shyy (2012). Effects of flexibil-ity on the aerodynamics of a hovering flexi-ble airfoil at reynolds number of 100 to 1000.AIAA Aerospace Sciences Meeting includingthe New Horizons Forum and Aerospace Ex-position.

Kassiotis, C., A. Ibrahimbegovic, R. Niekamp,and H. Matthies (2011). Nonlinear fluid-structure interaction problem. part i: im-plicit partitioned algorithm, nonlinear sta-bility proof and validation examples. HALarchives-ouvertes.fr.

Kuttler, U. and W. Wall (2008). Fixed-pointfluid-structure interaction solvers with dy-namic relaxation. Computational Mechan-ics, 43(1):6172.

13

Lohner, R. (2001). Applied CFD Techniques.John Wiley and Sons.

Maza, M., F. Flores, and S. Preidikman (2012).Interaccion fluido-estructura inestacionaria yno-lineal, con modelos de flujo potencial yestructuras de vigas. Asociacion Argentinade Mecanica Computacional XXXI, 771–795.

Naderi, A., M. Mojtahedpoor, and A. Beiki(2016). Numerical investigation of non-stationary parameters on effective phenom-ena of a pitching airfoil at low reynoldsnumber. Journal of Applied Fluid Mechan-ics 9(2), 643–651.

Olivier, M. (2010). A Fluid-Structure Interac-tion Partitioned Algorithm Applied to Flexi-ble Flapping Wing Propulsion. Ph. D. thesis,Universit Laval, Qubec, Canada.

Pedro, G., A. Suleman, and N. Djilali (2003).A numerical study of the propulsive effi-ciency of a flapping hydrofoil. InternationalJournal for numerical methods in fluids 42,493526.

Principe, J. and R. Codina (2009). On the sta-bilization parameter in the subgrid scale ap-proximation of scalar convection-diffusion-reaction equations on distorted meshes. In-ternational Center for Numerical Methods inEngineering.

Radmanesh, M., O. Nematollahi, M. Nili-Ahmadabadi, and M. Hassanalian (2014). Anovel strategy for designing and manufac-turing a fixed wing mav for the purpose ofincreasing maneuverability and stability inlongitudinal axis. Journal of Applied FluidMechanics 7(3), 435–446.

Taylor, G., R. Nudds, and A. Thomas (2003).Flying and swimming animals cruise at astrouhal number tuned for high power effi-ciency. Letters to Nature 425, 707711.

Triantafyllou, G. S., M. S. Triantafyllou, andM. A. Grosenbaugh (1993). Optimal thrustdevelopment in oscillating foils with appli-cation to fish propulsion. Fluids and Struc-tures 7, 205224.

Vazquez, J. G. V. (2007). Nonlinear Analysis ofOrthotropic Membrane and Shell StructuresIncluding Fluid-Structure Interaction. Ph. D.thesis, Universitat politecnica de catalunya,Barcelona, Spain.

Wall, W. A. and E. Ramm (1998). Fluid-structure interaction based upon a sta-bilized (ale) finite element method. IVWorld Congress on Computational Mechan-ics, Barcelona.

Wright, J. R. and J. E. Cooper (2007). Introduc-tion to aircraft aeroelasticity and loads. JohnWiley and Sons.

Zhu, Q. (2007). Numerical simulation of a flap-ping foil with chordwise or spanwise flexi-bility. AIAA 45(10), 2448–2457.

14