panacm 2015 paper

1st Pan-American Congress on Computational Mechanics - PANACM 2015XI Argentine Congress on Computational Mechanics - MECOM 2015

S. Idelsohn, V. Sonzogni, A. Coutinho, M. Cruchaga, A. Lew & M. Cerrolaza (Eds)

STEADY AND UNSTEADY ANALYSIS OFAERODYNAMICS WING SECTIONS AT ULTRA-LOW

REYNOLDS NUMBERS (RE < 10000)

DINO P. ANTONELLI1, CARLOS G. SACCO2 AND JOSE P. TAMAGNO3

1Universidad Nacional de Cordoba y CONICETAv. Velez Sarfield 1611, PC 5000 Cordoba, Argentina

[email protected]

2Instituto Universitario AeronauticoAv. Fuerza Aerea S/n, PC 5000 Cordoba, Argentina

[email protected]

3Universidad Nacional de CordobaAv. Velez Sarfield 1611, PC 5000 Cordoba, Argentina

[email protected]

Key words: aerodynamic wing sections, ultra-low Reynolds, CFD, steady and unsteadyflows.

Abstract. The purpose of this study is to describe phenomena that manifest them-selves in flows where Reynolds numbers are ultra-low (Re < 10000). To accomplishthis study, mathematical techniques capable of solving the Navier-Stokes equations forlaminar-incompressible flows are used. It is noted that a solver based on the FiniteElement Method provides an appropriate resolution procedure, however, it must also benoted that because of the incompressible assumption the character of the continuity equa-tion goes from hyperbolic to elliptic. Because of this, a Fractional Step method whichevolves toward a semi-implicit temporal integrator is used, and to handle the convectiveand pressure terms the so called Orthogonal Sub-grid Scale(OSS) algorithm is applied. Inaddition, the motion of the finite elements computational mesh through solving the Pois-son equation and optimizing each element metric, is implemented. Basic useful resultsdescribing the behavior of several 2D geometries at steady ultra-low Reynolds flows, arepresented. Different geometric parameters like thickness ratio, mean lines camber, shapeof leading edge, etc. are changed and its effects evaluated. Flow detachment featuresand their impact on main aerodynamic properties are assessed. Wing sections performingtypical unsteady flights like heaving, pitching, flapping and hovering are also analyzed,and its aerodynamic properties in terms of Strouhal numbers, reduced frequencies andReynolds numbers determined.

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Dino P. Antonelli, Carlos G. Sacco and Jose P. Tamagno

1 INTRODUCTION

The importance of ultra-low Reynolds flows lies in technology applications like MAVs(Micro Air Vehicles). Numerous cases require a deep understanding of the present phe-nomena in both steady and unsteady flights, to obtain maximum propulsive and handlingefficiencies. Basically due to Reynolds number effects, aerodynamic characteristics suchas lift, drag and thrust of a flight vehicle change considerably between MAVs and con-ventional manned air vehicles. In fact, in the nature, birds or insects flap their wingsinteracting with the surrounding air to generate lift to stay aloft or producing thrust tofly forward. The main powered flights are: flapping (flight with free stream) and hovering(flight without free stream).

Much research in this broad area have been made. The most significant that canbe named are: Kunz [8] in his thesis studied the behavior of different geometries insteady flows at ultra-low Reynolds number; Guerrero [7] carried out unsteady aerodynamicstudies at ultra-low Reynolds in 2D and 3D configurations built using the NACA 0012wing section; Pedro et al [10] with the purpose of studying the propulsive efficiency offlapping hydro-foil NACA 0012 at Re = 1100, (flow density ρ = 1kg/m3 and dynamicviscosity µ = 0.01kg(m.s) ), also carried out numerical simulations. A Finite VolumeTechnique with an additional equation for the pressure, an explicit temporal scheme andstructured grid, were used.

Ranges of non-dimensional numbers found relevant to unsteady flights of biological”flappers”, are also considered valid for MAVs. A characteristic one for flapping motionsis the Strouhal number St = 2fhha/U , where fh is the frequency and ha the amplitude.Therefore, the Strouhal number expresses de ratio between the flapping wing velocityand the reference velocity U . The reduced frequency given bay k = πfc/U is anotherparameter that can be interpreted as a measure of unsteadiness comparing the wave lengthof the flow disturbance to the chord c.

2D unsteady flow sinusoidal kinematics are characterized by the equations:

h(t) = hasin(2πfht+ φh) (1)

α(t) = αasin(2πfαt+ φα) (2)

where φh and φα are the phases angles.

2 NUMERICAL SIMULATION

2.1 Governing equations

The two-dimensional time-dependent Navier-Stokes equations are solved using the fi-nite element method, assuming incompressible-laminar flow which is justified since theMach number of MAV flight is M << 0.3 and the Reynolds number Re < 10000. Con-servation of mass and momentum are described by:

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∇.u = 0 in Ω× (t0, tf ) (3)

∂(u)

∂t+ (u.∇)u = −1

ρ∇p+ ν∇2u + fe in Ω× (t0, tf ) (4)

where Ω represents the analisys domain with boundaries Γu⋃

Γσ, whereas (t0, tf ) is timeinterval of analisys. The u is the two-dimensional flow velocity vector, ρ the constantdensity, ν the kinematic viscosity and p the pressure.

To represent the unsteady flow, eqs. 3 and 4 are solved in a fixed inertial referenceframe incorporating a moving mesh following the Arbitrary Lagrangian Eulerian (ALE)formulation [6]. This method combines the advantages of both the Lagrangian and Eu-lerian approaches. In the Lagrangian approach the computational mesh is moved suchthat the nodes follow material particles during motion. The computational mesh is fixedand the fluid moves with respect to the mesh. The ALE method incorporates a movingmesh using the Lagrangian method, where the mesh follows the motion of the geometryboundary, whereas the equations are solved using the Eulerian approach.

In order to obtain the ALE equations the velocity u in the convective term of themomentum equation needs to account for the mesh motion. Therefore the velocity of themesh um is subtracted from the flow velocity in the convective term. Then the Navier-Stokes equations in ALE formulation are obtained by:

∇.u = 0 (5)

∂(u)

∂t+ (c.∇)u +

1

ρ∇p− ν∇2u− fe = 0 (6)

where c = u − um is the convective velocity that represent the difference between fluidvelocity and mesh velocity.

In the present work, the algoritm of mesh movement is based in operations of opti-mization of smoothing, developed to the R©ANSYS software package [3].

2.2 Fractional Step algorithm

The equations previously presented can’t be solved by a numerical standard form be-cause incompressibility gives raise to a flow field restriction. There are several algorithmsto deal with this difficulty and the Fractional Step method is one of them. The methodmeets the LBB condition through the use of same order of approximation for velocity andpressure.

To apply the Fractional Step algorithm the momentum equation is divided in two parts:

un+1 = un + δt

[un+θ.∇un+θ + γ

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ρ∇pn − ν∇nun+θ + fn+θ

](7)

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un+1 = un+1 − δt

ρ(∇pn+1 − γ∇pn) (8)

and in the last two equations a new variable u known as fractionary momentum, isintroduced. If in eq. 8 the divergence is taken and the continuity equation is applied,results:

∇2(pn+1 − γpn) =ρ

δt∇.un+1 (9)

Through this equation the pressure is calculated. In addition, γ is a numerical parametersuch that its values of interest are 0 and 1. The θ parameter determine the kind oftemporal approximation.

2.3 Discret form of equations

The Finite Element Method is used to discretize the govern equations and provides anappropriate resolution procedure [9]. The resultant scheme is of first order (γ = 0) andthe temporal discretization (θ = 0) results in Euler forward. The test functions (ψh, φh)∈ Ψh × Φh are used such as 1:

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δt(un+1

h ,ψh) =1

δt(unh,ψh)− (unh.∇unh,ψh)− ν(∇unh,∇ψh)− (fne ,ψh) (10)

(∇pn+1h ,∇φh) =

ρ

δt

[(un+1

h − unh,∇φh)− (∇unh, φh)]

(11)

(un+1h ,ψh) = (unh,ψh)−

δt

ρ(∇pn+1

h , φh) (12)

The last equations system is semi-implicit because eqs. 10 and 12 are explicit (lumpedmass matrix) and eq. 11 for the pressure computation is implicit.

2.4 Stabilized scheme

The discretization of convective terms yields numerical instabilities, therefore stabiliza-tion methods must be used. In this work the Orthogonal Subgrid Scale (OSS) algorithmis applied [4],[5],[11]. The expresion for the convective stabilization term is:

STBu = τ1(unh.∇unh − πnh ,unh∇ψh) (13)

where πnh is the convective term proyection and it is defined in eq. 18. This equationadd to momentum eq. 10 and it is evaluated in tn, therefore it remains explicit.

The term stabilization of pressure to be added to the eq. 11 is:

1The notation used in the equations mean: (a,b) =∫a.bdΩ

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STBp = −(τ2(∇pn − ξnh),∇φh) (14)

where ξnh is the gradient pressure term proyection and it is defined in eq. 19. In addition,it is evaluated in tn, therefore it remains explicit.

The complete stabilized scheme is obtained:

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δt(un+1

h ,ψh) =1

δt(unh,ψh)− (unh · ∇unh,ψh)− ν(∇unh,∇ψh)− (fne ,ψh)− (15)

−(τ1(unh · ∇unh − πnh),∇unh · ∇ψh)

(∇pn+1h ,∇φh) =

ρ

δt+ τ2

[(un+1

h − unh,∇φh)− (∇unh, φh)]

+τ2

δt+ τ2(∇ξnh ,∇φh) (16)

(un+1h ,ψh) = (unh,ψh)−

δt

ρ(∇pn+1

h , φh) (17)

(πnh , ψh) = (unh.∇unh, ψh) (18)

(ξnh , ψh) = (∇pnh, ψh) (19)

where ψh ∈ Ψh. The system of equations of eqs. 15, 17, 18, 19 are solve in explicitform with lumped mass matrix and the system resultant of eq. 16 is solve in explicit formthrough of conjugate gradients with diagonal pre-conditioner.

It is noted that the formulation of the scheme isn’t in the ALE framework. To ac-count the mesh velocity is necessary introduce in convective and stabilizations terms, theconvective velocity c.

Finally the boundary conditons in viscous tensor and velocity are:

• Imposed velocity: u = uc

• No slip: u = 0

• No traction: n.σ.n = 0

3 VERIFICATION OF NUMERICAL CODE

For the cases studies included in Table 1 the following parameters are considered:pitching and heaving frequencies fα = fh = 0.225[Hz], reducy frequency k = 0.7096,maximum heaving amplitude ha = 1, phase angle ϕ = 90, Strouhal number St = 0.45and the variable parameter is the pitching amplitude.

From Table 1 it can be concluded that the results obtained in this work compare wellwith those given by [10] and [7], up to αa = 15, but no so much for greater anglesαa = 20 and αa = 25. In this cases computed values from this work tend to overpredictresults given by the other autors. Specific investigations about the reason for differenceswere not made.

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Pedro et al.[10] Guerrero[7] Present Workαa ct CLmax ct CLmax ct CLmax5 0.4324 8.3333 0.4245 8.0828 0.4311 8.207810 0.6511 7.4834 0.6576 7.1699 0.6556 7.240015 0.8226 6.6307 0.8360 6.5435 0.8246 6.390420 0.9337 5.8176 0.9389 6.1133 0.9960 5.511325 1.0046 5.0558 0.9601 5.6080 1.0900 4.9910

Table 1: Average thrust ct and maximum lift CLmax coefficients. Comparison for flapping NACA 0012cases between present work, Refs. [10] and [7].

4 RESULTS AND DISCUSSION

4.1 Steady analisys of aerodynamic airfoils

Figure 1: Lift and drag coefficients for thickness ratios and camber ratios over standart NACA 4 digitairfoils. (a) and (b) thickness ratio effects. (c) and (d) camber ratio effects. (The inviscid curve is from[8]).

To analize how the thickness ratio behaves in ultra-low Reynolds flow, comparisonsbetween four digits symetric NACA airfoils (0002, 0006, 0008) at two Reynolds numbers(Re = 2000 and Re = 6000) are carried out. Results are presented in Fig. 1.a and 1.b.Consider first the 0006 and 0008 thickness airfoils. In the quasi-linear ideal range of angles

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of attack (0− 6) for a symmetric airfoil (Fig.1.a), a very important reduction in lift canbe observed. If the Reynolds number is decreased the lift slope tends to improve, but suchimprovement is comparatively less than the decrease due to thickness. However, in NACA0002 airfoil the lift slope becomes greater when the Reynolds is higher (Re = 6000). Itappear that his behavior is related to a lower leading edge suction peak and subsequentdelayed stall. Note from Fig. 1.a that as the thickness ratio get smaller, the closerthe viscous results get to the ideal values. Fig. ??.b shows the strong increase of dragcoefficients as the Reynolds number decreases.

To analize the camber behavior a comparison between NACA 2302, 4302 and 6302 isperformed. Lift results of numerical simulations plotted in terms of α− α0, being α0 thezero lift angle, are shown in Fig. 1.c. The 6302 airfoil lift slope, is 30% greater than 2302airfoil. If CL = 0, note the large increment of CD as the camber increases Fig. 1.d, it canbe attributed to leading edge early detachments.

4.2 Unsteady analisys of aerodynamic airfoils

Figure 2: Average thrust coefficient ct and propulsive efficiency η for NACA airfoils (0004, 0006 and0012) in heaving motion. Plots (a) and (b) apply to f = 1[Hz]. Plots (c) and (d) apply to f = 2[Hz].

The fundamental parameter of unsteady analysis is the Strouhal number, defined asSt = 2fha/U . Taylor et al. [12] and Triantafyllou et al. [13] performed a study of wingfrequencies and amplitudes, and cruise speeds across a range of birds, insects, fishes andcetaceans, to determine Strouhal numbers in “cruising” flight. They found 75% of the 42

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species considered fall within a narrow range of 0.19 < St < 0.41 [7]. Therefore, a similarrange of Strouhal numbers has in this work been selected.

4.2.1 Heaving

The kinematic of heaving motion is given by eq. 1. The analysis is applied to NACA’sfour digits (0004, 0006 and 0012) symmetrical airfoils. The kinematics parameters are:two values of heaving frequencies fh and a variable Strouhal number throughout theheaving amplitude ha and Re = 1100. The average thrust coefficient ct and propulsiveefficiency η = ct/cp (where cp is the power coefficient input), are in terms of the Strouhalnumber presented for fh = 1[Hz] and fh = 2[Hz] in Fig. 2.a and 2.b, and in Fig. 2c and2d respectively. Note the numerical results obtained by Guerrero [7] in Fig. 2.a and 2.b.

Figure 3: Comparision at differents times of velocity contours between NACA 0004 and NACA 0012airfoils (fh = 1[Hz] and St = 0.3).Times (a) and (e) t = 0.45[s], (b) and (f) t = 0.86[s], (c) and (g)t = 1.29[s], (d) and (h) t = 1.64[s].

The flow motion topology expressed by Figure 3, helps to understand the results ofthe simulations. Therefore, comparisons at different times of velocity contours betweenNACA 0004 (Fig. 3 a,b,c,d) and NACA 0012 (Fig. 3 e,f,g,h) are shown. The formationof leading edge vortices (LEV) and its convection into the wake, can be observed [1].

4.2.2 Flapping

The name flapping is applicable to a combined motion of heaving and pitching, conse-quently the kinematics relations given by equations 1 and 2 are simultaneously applied.

In Fig. 4a and 4b are shown average thrust coefficients ct and propulsive efficienciesη applicable to NACA symetric airfoils 0004 and 0012, as function of the pitching angle

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5 < αa < 25 with fα = fh = 0.3, ha = 0.5[m], ϕ = π/2 and St = 0.3. In Fig. 4c and 4dare shown average thrust coefficients ct and propulsive efficiencies η applicable to NACAsymmetric airfoils 0004 and 0012, as function of the heaving amplitude 0.025 < ha < 0.5(0.05 < St < 1) with fh = fα = 1, αa = 15 and ϕ = π/2.

Figure 4: Average thrust coefficient ct and propulsive efficiency η in heaving motion for thickness ratiovariation in NACA 0004, 0012 at Re = 1100. (a) and (b) pitching amplitude variable αa. (c) and (d)heaving amplitude variable ha.

4.2.3 Hovering

The kinematics relations in hovering flight are given by equations 1 and 2 simulta-neously applied. The kinematic parameters utilized in the simulation of hovering are:ha = 0.5, fα = 0.75, fh = 0.75 and ϕ = π/2 and the Reynolds number is defined byRe = 2fhρπhac/µ because the free stream velocity is null. Average lift coefficients cland η efficiencies were obtained simulating the hovering of a NACA 0012 airfoil, and areplotted as function of the Reynolds number in Figures 5a and 5b. The Reynolds numbercovers the range 100 < Re < 1000.

On the other hand the wake topology is studied in Fig. 6. It can be observed theLEV and TEV (trailing edge vortex) at Re = 150, αa = 20 and ϕ = π/2 , causedby some stroke and its motions indicated by arrows. Three typical mechanism neededto understand the behavior of hovering are present: wake capture and diffusion effects,dynamic stall and roll-up effects.

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Figure 5: Hovering motion over NACA 0012 airfoil with ha = 0.5, fα = 0.75, fh = 0.75 and ϕ = π/2.(a) average lift coefficient cl. (b) efficiency η hovering motion for two pitching amplitudes.

Figure 6: Velocity contours in hovering motion to NACA 0012 at Re = 150, αa = 20 and ϕ = π/2. (a)t = 0.6s, (b) t = 0.92s, (c) t = 1.20s, (d) t = 1.64s, (e) t = 2.00s, (f) t = 2.26s. (LEV C and TEV C arethe previous stroke vortexes to capture and LEV N the new vortex generated).

5 CONCLUSIONS AND FUTURE WORK

What has been sought with this work is the confirmation that current computationalknowledge can to the ultra-low Reynolds number applications (Re < 10000), be extended.The main assumptions made about the flow field are: two dimensional incompressible,fully laminar steady and unsteady flows. The fully laminar assumption is the most phys-ically accurate in the range of Reynolds numbers and angles of attack of interest here.

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5.1 Steady Analysis

Two dimensional analysis allows a broad spectrum of parameters to be considered andprovides a baseline for future more detailed studies. The geometric parameters investi-gated included thickness, camber etc, and it was intended to address the fundamentalquestion of whether section geometry is still important at ultra-low Reynolds number.The most obvious effect of operating at ultra-low Reynolds numbers is a large increasein the section drag coefficient, however, the increase in drag is not reciprocated in lift,resulting in a large reduction in the L/D.

A ultra-low Reynolds numbers flow, is dominated by viscosity and the so called Bound-ary Layer concept is no longer applicable. Here it is generalized as the lower velocityviscous flow region adjacent to the body over which the pressure gradient normal to thesurface is almost null. The extended constant pressure from the surface implies that thewing section effective geometry is significantly altered. As a result the pressure recoveryis reduced and besides impacting on drag, at positive angles of attack the large effect is onlift. Viscous effects in thin wing sections thickness, significantly reduce the leading edgesuction peak and the associated reduction in slope of the adverse pressure recovery, delaysthe onset of the stall. It can be stated that leading edge separation is delayed in thinwing sections and trailing edge separation is delayed in thicker sections. This behaviourcould be potentially beneficial to lifting performance.

The effects of camber do not differ significantly from those observed at much higherReynolds numbers. The fact that as the Reynolds numbers and section maximum thick-ness are reduced the details of the thickness distribution becomes less relevant, it allowsto conclude that the camber-line is the dominant factor in performances.

5.2 Unsteady analysis

The highlight of the Finite Element software here used, is the ability to create mobilegrids needed to simulate unsteady flights like heaving, flapping and hovering.

Symmetric wing sections are considered in studying the heaving motion. Average thrustcoefficients and propulsion efficiencies are computed for given motion frequencies, and areplotted in terms of a Strouhal number determined using the amplitude of heaving. As ahelp for understanding the simulation results, figures are shown where velocity contoursfor two wing sections are compared at different times. The generation and displacementof vortices as the wing section executes the heaving motion, are well described.

Combinations of pitching and heaving motions (flapping) were simulated for symmetricwing sections, and thrust coefficients and propulsive efficiencies determined. Maximumpitch and vertical displacement amplitudes were taken as variables of plotting.

Hovering, is perhaps the type of flight that capture the greatest interest in the develop-ment of MAV. Average lift coefficients and power efficiencies are obtained simulating theflight of a symmetric wing section and the results plotted in terms of an ad-hoc Reynoldsnumber defined taking into account that there is no free-stream. Classical aspects of

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the hovering flight (leading and trailing edge already generated vortices, its capture anddiffusion by the wake, and corresponding new generation), are shown.

5.3 Future area of research

So far, the fluid dynamic studies have been conducted and applied to a rigid 2D model.It is intended to extend first, the ultra-low Reynolds number area of research to 3D rigidfinite span lifting wings, to the development of the viscous flow region adjacent to thesurface, to describe stall patterns and wakes coming forth. Later, the rigid wing will bereplaced by an elastic model and the coupling fluid-structure accounted for.

REFERENCES

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[8] Kunz, P. J. Aerodynamics and desing for ultra-low Reynolds number flight . PhdThesis. Stanford University (2003).

[9] Lohner R. Applied CFD Techniques. John Wiley and Sons (2001).

[10] Pedro, G. Suleman, A. and Djilali, N. A numerical study of the propulsive efficiencyof a flapping hydrofoil. International Journal for numerical methods in fluids (2003)42:493–526.

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[11] Principe, J. and Codina, R. On the stabilization parameter in the subgrid scaleapproximation of scalar convectiondiffusionreaction equations on distorted meshes.Tech. rep., International Center for Numerical Methods in Engineering. (2009).

[12] Taylor, G., Nudds, R., and Thomas, A. Flying and swimming animals cruise at astrouhal number tuned for high power efficiency. Letters to Nature (2003) 427:707–711.

[13] Triantafyllou, G. S., Triantafyllou, M. S. and Grosenbaugh, M. A. Optimal thrustdevelopment in oscillating foils with application to fish propulsion. Fluids and Struc-tures. (1993) 7:205–224.

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