Validation of Unsteady Reynolds-Averaged NavierStokesSimulations on Three-Dimensional Flapping Wings

Stephan E. Bansmer and Rolf Radespiel

Technische Universitt Braunschweig, 38106 Braunschweig, Germany

DOI: 10.2514/1.J051226

A combined experimental and computational study is presented for a wing segment undergoing a combined

pitching, plunging, and rolling motion at Reynolds number of 100,000, where transition takes place along laminar

separation bubbles. The numerical simulation approach addresses unsteady Reynolds-averaged NavierStokes

solutions and covers three-dimensional transition prediction for unsteadymeanows.Thenumerical simulations are

validated using high-resolution, phase-locked stereoscopic particle image velocimetry for a three-dimensional

apping casewith a reduced frequency of k 0:25. Theow reveals strong unsteadiness resulting inmoving laminarseparation bubbles, whose spatial extensions are varying in the spanwise direction. The experimental results are

well captured by the numerical simulations performed in this study. Because of the vortex structure in the wake

of the wing segment, the three-dimensional aerodynamics cannot be reproduced as a spanwise sequence of

two-dimensional results.

Nomenclature

A = amplitude of wave disturbance oscillation, mc = chord length, mcd = drag coefcient, sum of cd;f and cd;pcd;f = friction drag coefcientcd;p = pressure drag coefcientcl = lift coefcientcm = vector of x, y, and z moment coefcientsct = thrust coefcientcx, cy, cz = force coefcientsf = frequency of the apping motion, Hzi = integer incrementj = imaginary unit, j 1pk = reduced frequencyM = number of samplesN N = factorRe = Reynolds numberT = period time of one apping cycle, st = time, sU1 = freestream velocity, m=su, v, w = velocity components, m=su0, v0, w0 = turbulence velocity components, m=sv = velocity vectorx, y, z = Cartesian coordinates, mx = vector of Cartesian coordinatesxt = Cartesian coordinate of the transition location, mx0 = Cartesian coordinate of the point of neutral

stability, mz^ = plunging amplitude, m = complex wave number in x direction, 1=meff = effective angle of attack, degi = imaginary part of the complex wave number in x

direction, 1=m = complex wave number in y direction, 1=meff = amplitude of the effective angle of attack, deg

= phase difference between pitch and plunge motions,deg

= induced angle of attack due to plunge motion, degp = propulsive efciencyL = pitch/plunge amplitude ratiot = eddy viscosity, kg=m s% = density, kg=m3

xz = turbulent shear stress, kg=ms2 = pitch angle, geometric angle of attack, deg_ = vector of angular pitch, roll, and yaw velocities^ = pitch amplitude, deg0 = mean pitch, mean geometric angle of attack, deg = roll angle, deg! = complex frequency of disturbance wave, Hz

I. Introduction

F LIGHT physics of birds intersect with some of the mostchallenging problems in modern aerospace engineering:unsteady three-dimensional separation, moving transition withlaminar separation bubbles in boundary layers, unsteady ightenvironment, aeroelasticity, and anisotropic wing structure are just afew examples [1].Focusing on three-dimensional aerodynamics, comparisons

between experimental and computational results of a nominallythree-dimensional apping-wing segment at low Reynolds numbers(Re 100; 000) are presented. The apping motion is prescribed bya combined plunging, pitching, and rolling motion with a phasedifference of 90 deg between plunging and pitching.In the past, Neef [2] investigated the propulsive efciency of a

apping NACA 0012 wing segment using different inviscid owsolvers. The kinematics of the wing were prescribed by combinedplunging, pitching, rolling, and torsional motions. The wingtipvorticity was identied as a major loss of propulsive efciency. Theeffect of the combined rolling and torsional motion was remarkable,because it enables the wing to generate thrust during upstroke anddownstroke of the apping cycle.However, the viscous effect in this low Reynolds number range of

100,000 cannot be neglected, since the laminarturbulent transitiontakes place along laminar separation bubbles (LSBs). Figure 1describes the physics of a LSB. The oncoming laminar boundarylayer separates, which is caused by a pressure increase along theairfoil contour. According to Spalart and Strelets [3] and Rist [4], theseparated ow performs the transition process from laminar toturbulent ow following a gradual development of theprimary instabilities from TollmienSchlichting instabilities toward

Received 15 February 2011; revision received 21 May 2011; accepted forpublication 27May 2011. Copyright 2011 by the authors. Published by theAmerican Institute of Aeronautics and Astronautics, Inc., with permission.Copies of this paper may be made for personal or internal use, on conditionthat the copier pay the $10.00 per-copy fee to theCopyright Clearance Center,Inc., 222RosewoodDrive, Danvers,MA01923; include the code 0001-1452/12 and $10.00 in correspondence with the CCC.

Research Associate, Institute of Fluid Mechanics, Bienroder Weg 3;s.bansmer@tu-braunschweig.de.

Head of the Institute, Institute of Fluid Mechanics, Bienroder Weg 3.Senior Member AIAA.

AIAA JOURNALVol. 50, No. 1, January 2012

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KelvinHelmholtz instabilities [5]. The resulting turbulentuctuations in the ow enhance momentum transport toward thewall, and the ow reattaches to the airfoil contour. The resultingregion of circulating ow is called the laminar separation bubble.LSBs are usually not desired in airfoil design, because they increasethe pressure drag of the airfoil due to a higher displacement-thicknesslevel of the boundary layer.In this contribution, a validated and efcient numerical tool will be

provided, which covers the unsteady ow case of a three-dimensional apping-wing segment at transitional low Reynoldsnumbers. This work is based upon previous two-dimensional results.In the past, the authors developed a validated method to predict theviscous ow around an oscillating airfoil performing a combinedpitching and plunging motion [6,7]. The ow solver of our choicewas FLOWer [8]. For the present three-dimensional investigations,the authors migrated toward the TAU code of the DLR, GermanAerospace Center [9], which includes an implemented three-dimensional transition prediction [10,11]. Therefore, the presentapproach consists of three steps:1) Generate a birdlikewing segment. Thework draws on naturally

evolved airfoils. Although technical airfoils might achieve a betteraerodynamic performance, naturally evolved airfoils are assumed tobe an optimum in the parameter space of lightweight design, aero-dynamic performance, and maneuverability. Therefore, the shape ofthe hand pinion of a seagull was used as design paradigm, alsobecause the hand pinion of bird wings is known to be the thrust-producing part [12], which is favorable for future micro air vehicledesign. Seagulls were used as an inspiration because, in contrast tostorks or hummingbirds, they use the apping ight at moderatereduced frequencies as their normal cruise mechanism. As a result,the SG04 airfoil was developed [13], which is extruded in thespanwise direction to a wing segment with an aspect ratio of 2.2) Verify the TAU code simulations. Previous two-dimensional

FLOWer results have to be reproduced with the TAU code (seeSec. V.A).3) The computational results of the TAU code have to be validated.

Therefore, a rigid SG04 wing segment was manufactured inlightweight design. High-resolution particle image velocimetry ofthe boundary layer was used to capture velocity elds and turbulent

shear-stress distributions around the apping-wing segment at areduced frequency of k 0:25.One feature of three-dimensional apping wings is the complex

vortex system in the wake (see Fig. 2). It is composed of asuperposition of start/stop eddies with wingtip vortices. In thiscontribution,wingtip vortices due to the nite span are suppressed byside walls at the end of the wing segment; however, a dynamicallyevolving vortex sheetwith spanwise variation of unsteady circulationis present.

II. Birdlike Wing Segment

A. Aerodynamic Design

Compared with conventional airfoils, two major design aspectscan be foundwhen examining the airfoil of a seagull in the vicinity ofits hand pinion: rst, a large maximum camber compared witharticial airfoils, and second, the position of maximum thickness islocated close to the leading edge (see also Fig. 3).There are several reasons why the position of maximum thickness

is situated near the leading edge. Considering the wing anatomy, theskeleton and muscles run in this section, whereas at the trailing-edgeregion of the airfoil, only the feathers determine the airfoil shape.From an aerodynamic point of view, there are advantages of thisdesign as well. On the one hand, the adverse pressure gradients alongthe upper surface can be kept reasonably small. This yields thinlaminar separation bubbles with low pressure drag losses. On theother hand, thin airfoils usually exhibit a small range of applicableangles of attack where no stall occurs. Airfoils with their position ofmaximum thickness in the vicinity of the leading edge exhibit anincreased nose radius, which results in a relatively large angle-of-attack range with attached ow. A large relative camber of 8% wasmeasured byBilo [14] using narcotized birds.However, observationsin nature revealed that this value is usually smaller for gliding ight(approximately 4%), although in wind-tunnel experiments withliving birds the maximum camber during one apping stroke wasfound to vary from 8 to 12%.Based on these design aspects, a new birdlike airfoil, the SG04,

was developed [13] (see Fig. 4). This prole represents the handpinion of thewing, which is known to be the thrust-producing part ofa apping wing [12], and corresponds in its shape to section 3 inFig. 3. Sections 4 and 5 of Fig. 3 were not used as an airfoil designinspiration, because these sections are only determined by the shapeof feather tips. Beyond, with respect to bending stiffness, such thinairfoils (t=c < 1%) are difcult to manufacture for validationexperiments in wind tunnels. SG04 has a maximum thickness and amaximum camber of 4%, where the maximum camber is located atx=c 40%. The aerodynamic design of the SG04 involved bothaerodynamic analysis and inverse design according to the bubble

Fig. 1 Sketch of a laminar separation bubble by Horton [30]

(corrected).

Fig. 2 Three-dimensional vortex system of a apping wing [32,33].

Fig. 3 Comparative investigations on airfoil shapes of birds from

Oehme [34]. Shown are ve idealized bird-wing sections.

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ramp approach introduced by Selig [15] to reduce the size of thelaminar separation bubble for a broad range of angles of attack.Figure 5 shows the drag polar of the SG04 airfoil computed byFLOWer, a validated unsteady Reynolds-averaged NavierStokes(URANS) solver [7], and XFOIL. The eN method was used fortransition prediction with the critical N factor of 10. This choicecorresponds to the ow quality at the test Reynolds number of100,000 in the wind tunnel, which was used for the validationexperiments (see also [6]). Although XFOIL is a very simple tool topredict steady airfoil aerodynamics, its polar is in good agreementwith the results of the more complex FLOWer solver. Additionally,the drag decompositions into friction and pressure drag are plotted,which are approximately in the same order of magnitude in this lowReynolds number range of 100,000. Considering the small pressuredrag values, the design goal of the SG04 airfoil with small laminarseparation bubbleswas attained. To be able to compare the polar witha SD7003 airfoil at an equivalent Reynolds number, experimentaldata from Selig et al. [16] is added to the chart. Mainly due to thehigher camber, the SG04 polar is shifted to higher lift values and hastherefore a smaller drag at the same lift performance. However, thedisadvantage of thin airfoils is clearly visible with a decreasedoperational range of lift.For the three-dimensional investigations in this contribution, the

SG04 airfoil was extruded in the spanwise direction to a wingsegment with an aspect ratio of 2.

B. Kinematics of the Oscillating-Wing Segment

In the two-dimensional case of an oscillating airfoil, the motion iscombined by a plunging zt=T motion and a pitching oscillationt=T around the quarter-chord for one apping period0 t=T < 1:

zt=T z^ cos2t=Tt=T ^ cos2t=T =2 0 (1)

Thesemotions inuence the local oncoming ow conditions in theframe of reference of the airfoil. For example, when the airfoil movesdownward, the airfoil in its frame of reference is exposed with anadditional ow component in the upward direction. Consequently,the effective oncoming ow vector changes direction and magnitude(see Fig. 6). This also has an impact on the local angle of attack. The

geometric angle of attack is superimposed by the induced angle ofattack resulting from the plunging motion of the airfoil, whichnally yields the effective angle of attack eff :

efft=T t=T t=T ^ cos2t=T =2 0

arctan

1U1

@zt=T@t

xc4

eff sin2t=T 0 (2)

The pitching motion affects similarly the oncoming ow vector.However, its induced ow velocity in the z direction is a function ofthe distance to the rotational axis of the pitching motion. Hence, thisinuence can be interpreted as a decambering of the airfoil.To generate three-dimensional wing kinematics, rolling motion

t=T ^ cos2t=T is superimposed in addition (see Fig. 7).Consequently, the effective angle of attack varies in the spanwisedirection. However, the three-dimensional effect of wingtip vorticesdue to the nite span is suppressed by side walls of the wind tunnel.The reduced frequency k is introduced, often given by

k fcU1

(3)

The inverse of k is ameasure how far the undisturbed air passes theairfoil during one apping cycle. Therefore, the reduced frequencycan be used to classify the level of aerodynamic unsteadiness. Forseagull ight conditions, k has values around 0.2. This can be derivedfrom cruising ight data of birds (see Pennycuick [17] and Herzog[18]). According to this data, a kelp gull (Larus dominicanus) has amean chord length of 0.16 m and a apping frequency of 3.46 Hz.Assuming a cruise speed of 8 m=s, the reduced frequency can bedetermined to be k 0:22. Insects are known to have a reducedfrequency of about 1.0 [19].

U0.000 0.010 0.020 0.030

0.4

0.8

1.2

XFOIL, N=10SD7003 (Selig et al. [16])FLOWer, N=10

V

G

B

C

C

Fig. 5 Drag polar of the SG04 airfoil [7]; Re 100; 000, steadyconditions, and additional SD7003 data [16].

Fig. 6 Change from geometric to effective angle of attack efft dueto the plunging motion _zt.

Fig. 4 Birdlike airfoil SG04.

Fig. 7 Oscillating SG04 wing segment. In the experiments, the wing

segment is limited by the wind-tunnel sidewalls. Consequently, wingtipvortices are suppressed. Caused by the roll motion t, the gap sizebetween wall and wingtip is continuously changing in the experiments,

which are presented later. A rubber gasket closes this gap.

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III. Experimental Investigations

The objective of the experimental investigations is to capture theoweld in the boundary layer of the oscillating-wing segment andits turbulent quantities, such as the turbulent shear stress. Themeasurements carried out serve as a three-dimensional validationcase and are not prescribing a database for parametric studies.

A. Wind Tunnel

The experiments were carried out at the low-speed low-noisewindtunnel (see Fig. 8). The inlet of the Eiffel-type tunnel is covered by aeece mat 30 mm in thickness. Afterward, the air passes astraightener made out of aluminum honeycombs, 14 mm in diameterand 200 mm in length and then nally through a ne-mesh wovenscreen. In the large settling chamber, small-scale turbulence isdissipated, and a Boerger-type nozzle contracts the air at a 16:1 ratio.Consequently, the air has a very low turbulence level in the 400 600 mm sized test section. The wind tunnel is driven by a 4 kW,acoustically encapsulated, speed-controlled, three-phase asynchro-nous motor, which produces stable wind-tunnel speeds from 2 up to20 m=s. The diffusor is mounted on a rail system, which allows oneto interchange modular test sections. The laboratory is lined withopen-celled acoustic foam.

B. Motion Apparatus and Wing Segment

To create a combined plunging and pitching motion of the airfoilas denoted by Eq. (1), a special apping-motion apparatus [13] wasused. The apparatus synchronizes pitching and plungingmotionwitha mechanical gearing, depicted in Fig. 9. A broad range of motionparameters can be adjusted: plunging amplitude from 0 to 0.1 m,pitching amplitude from 0 to 25 deg, and apping frequency from 0to 10 Hz. The rolling motion of the three-dimensional oscillating-wing segment is created by different plunging amplitudes at bothwingtips (again see Fig. 7). Caused by the rolling motion t, thegap size between wall and wingtip is continuously changing. Arubber gasket closes this gap. Its smooth shape protects the wingtip

against ow separation and avoids the generation of wingtip vortices(see Fig. 10). Additionally, a light barrier is mounted at the rig,enabling the connected measurement systems to be triggeredaccording to the operated apping frequency. The estimated motionaccuracy for the validation case is about 0.8 mm for the plungingmotion and 0.2 deg for the pitching motion. These values weremeasured by capturing the airfoil position at a constant phase for 500apping cycles and determining its standard deviation.The SG04 wing segment used for the experimental investigations

wasmade of a carbon-fabric shell, which was reinforced by a closed-cell rigid foam. For a high bending stiffness, a carbon-ber spar wasintegrated. A top coat of polyester resin provided a smooth surface.The mass of the wing segment is 360 g.

C. Particle Image Velocimetry Measurements

The stereoscopic particle image velocimetry (PIV) was chosen tocapture the oweld in the boundary layer of the apping airfoil. Theadvantage of this nonintrusive measurement technique is not only togather a complete velocity eld instead of pointwise information, butalso to obtain quantitative data with reasonable accuracy [20]. Thedetailed setup is sketched in Fig. 11 as a top view of the wind-tunneltest section. The oncoming ow from left passes the moving airfoil,which is driven by the motion apparatus outside the test section. Adouble-pulsed Nd:YAG laser (Quantel Brilliant, energy: 2150 mJ, wavelength: 532 nm) mounted on the top of the windtunnel creates a thin light sheet in the chordwise direction. To capturethe three-dimensional aerodynamics around the wing segment, thelight sheet was positioned consecutively at three different spanwiselocations. Two LaVision Imager Intense cameras capture images ofthe illuminated tracer particles (oil particles with a mean diameter ofabout 1 m). The stereoscopic setup was necessary because theapping-motion apparatus does not allow for direct visible accessnormal to the laser light sheet, i.e., no standard PIV was possible. Itshould be mentioned that the eld of view of the cameras was only2:5 2 cm2; otherwise, there would not be sufcient resolution tocapture the boundary-layer ow (see also Fig. 12). These dimensions

Fig. 8 Schematic of the low-speed low-noise wind tunnel.

a) Schematic sketch b) Setup at wind tunnelFig. 9 Flapping-motion apparatus [13,31].

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are very small compared with the airfoil chord length of 20 cm.Hence, the camera system had to be attached to a translation device inorder to move the system in the plunging and chordwise directionswithout changing the alignment of the cameras. The light-sheetthickness was adjusted to a value of about 2 mm. This allows tomeasure the out-of-plane velocity with the stereoscopic setup. Theshort distance between light sheet and camera of about 0.55 menabled the camera lens to focus on the light sheet with a smallbandwidth of optical depth. Therefore, it can be ensured that onlyparticles within the light sheet are captured in the particle images. Tocapture the oweld at a constant phase angle, phase-locked imagingwas performed. For this purpose, a transistortransistor logic signalfrom the apping-motion apparatus, which triggers at the beginningof each apping cycle, was captured and shifted in time by a Stanforddelay generator DG 535. When the delayed signal was detected bythe programmable timing unit (PTU9 by LaVision), the laser ashand the camera exposure were initiated. The PIV system wascontrolled by theDavis Software of LaVision. The sampling ratewasadjusted to one particle image pair per apping cycle.

Once the particle image acquisition of at least 500 image pairs foreach of the independently captured measurement windows wascompleted, the velocity vector eld of the ow around the airfoil andits turbulent quantities were determined. First, a wobble correctionwas performed as follows: When the laser light sheet touched theairfoil surface, the so-called reection line was visible on the cameraimages. The thickness of this reection line varied from 3 to40 pixels, depending on the local curvature of the airfoil. Because ofthe phase-locked imaging, this reection (which indicates theposition of the airfoil) should be always at the same location.However, the reection line is wobbling about fractions of 1 mm inthe camera images. This wobbling had to be removed: for the laterensemble-averaging procedure of the vector elds (to compute themean ow vector eld, etc.), the airfoil has to be at the same position.The wobble correction is composed of three steps:1) Transform the image from the camera coordinate system into

the world coordinate system. Now, the four particle images of onecapturing cycle (two cameras, each capturing at two moments t andtt: Pt1,Ptt1 , Pt2, and Ptt2 ) are given in the same coordinatesystem. In consequence, the correlation procedure, which isdescribed in the next step, has to be performed only once.2) A distinctive area of the reection line, which can be found on

each of the 500 image pairs, has to be localized. This area is markedon the rst of 500 images Pt1jimage1 and will be correlated with theremaining 499 images Pt1jimagej (see Fig. 13). The resulting 499displacement vectors are used to move all 499 images to the origin ofimage one. The imagesPtt1 ,P

t2, andP

tt2 are treatedwith the same

displacement vectors.3) Retransform the corrected images from the world coordinate

system into the camera coordinate system.The presented procedure can only remove translational wobbling,

angular errors of the reection line are not corrected. This could bedone by correlating two points on the reection line instead of one.However, it is the experience of the authors that angular correctionsdiminish the particle image accuracy due to unavoidable subpixelinterpolation. Having performed several image preprocessingtechniques to improve the particle image quality, the particle dis-placement evaluation was done in a next step using a crosscorrelation scheme. The reection line was entirely masked out toavoid correlation errors near the wing surface. A multipassinterrogation scheme with decreasing interrogation window size(from 128 128 pixels down to 32 32 pixels), 50% overlap, andelliptical weighting function was applied. Based upon themeasurements presented in the following sections, the measurementuncertainty for the ow velocity in the boundary layer can beevaluated exemplarily by a scheme of Raffel et al. [20] (see Table 1).The scheme itself was derived from Monte Carlo simulations witharticial particle images. Consequently, each component of therandom error (for instance, the inuence of the particle diameter) canbe investigated individually. This yields a velocity uncertainty in theboundary layer due to random effects of 0:058 m=s.The resulting set of at least 500 vector elds for eachmeasurement

window was postprocessed afterward. This was mandatory to lterout nonphysical vectors, which would impair the results of theensemble-averaging procedure. Ensemble-averaging is the statisticaltask to compute the mean velocity eld, as given by the equation

hvx; ti hux; tihvx; tihwx; ti

0@

1A 1

M

XMi1

vix; t

Fig. 10 Sketch of the rubber-gasket installation (not true to scale).

Fig. 11 Stereoscopic PIV setup as a top view.

x/c

z/c

0 0.2 0.4 0.6 0.8 1

0

0.1

Fig. 12 Eleven measurement windows were used to capture theboundary layer of the apping SG04 airfoil.

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Moreover, the turbulent shear stress xz can be computed [21]:

xz % hu0 w0i % 1MXMi1

ui huiwi hwi (4)

Finally, all measurement windows per investigated phase anglewere placed at the correct position on the airfoil to obtain thecomplete distribution of the computed quantities along the airfoilsurface. This operation was done with TECPLOT 360.

IV. Numerical Simulation of theOscillating-Wing Segment

The numerical simulation approach is based onURANS solutions,which are coupledwith a sound transition-predictionmethod.Detailsof this approach can be found inRadespiel et al. [6] andKrimmelbeinet al. [10,11]. This section only focuses on a brief overview of thismethodology.

A. Simulation Approach

The time-dependent mean ow was computed by solving theunsteady Reynolds-averaged NavierStokes equations. Linearstability analysis, which directly investigates the velocity prolesof the Reynolds-averaged NavierStokes solution was then applied[10,21]. Waves due to TollmienSchlichting and KelvinHelmholtzinstabilities were predicted by the stability analysis [4]. Theamplication rates were then used to predict the transition locationusing an integration scheme for mode amplitude ratios that takesunsteady ow effects into account. The largest amplitude exponentswere nally compared with a critical N factor in order to determinethe transition location.

B. NavierStokes Code TAU

The NavierStokes solver TAU [9] requires unstructured meshes;its discretization scheme is a nite volume approach. A second-order-accurate central-difference scheme with scalar dissipationwas applied to evaluate convective uxes. Local-time-stepping,preconditioning, and multigrid operations were performed toaccelerate the computation. A second-order-accurate implicit dual-time-stepping scheme was used for the time-accurate computations.The Menter two-layer k-model [22] was chosen for the turbulencemodeling.

C. Linear Stability Analysis with LILO

To quantify the local amplication of disturbances, the stabilityequations of the laminar boundary layer are solved. LILO [23] treatslaminar compressible boundary layers; here, its option of assumingparallel ow is used. The harmonic-wave assumption is applied tothe velocity components u, v, w as well as for static pressure p andtemperature T, given exemplarily by the relation

q ~qx; y; z; t q0x; y; z; t;q0x; y; z; t q^ expjx y !t (5)

Assuming the frequency of the temporal distribution of the basicow state is much smaller than the frequency of the single wavemode, the time dependence of the evolutionmatrices of the system ofve stability equations vanishes. Consequently, the stability problemcan be solved at a discrete time. LILO computes the complexeigenvalue ! of the temporal stability problem in which the user hasto specify the real wave numbers and for 3-D cases. LILO is alsoable to detect crossow and attachment-line instabilities. However,in the present validation case, these transition mechanisms did notoccur, neither in the experimental nor in the computational results.

D. Transition Prediction

In many cases, the location of the nal laminar breakdown toturbulence is dominated by the behavior of the primary instabilitieswith their exponential growth. In these cases the point where theboundary layer becomes fully turbulent correlates well with a certainamplication factor of the most unstable primary wave that iscalculated from the point of neutral stability x0. These ndingsconstitute the eN method [24]. A suitable mathematical formulationof this method can be derived from the kinematic wave theory [25].Disturbance waves with discrete wave frequencies are assumed totravel downstream and to grow exponentially. Having calculated theamplication of these waves,

Ax A0 expZ

x

x0

i dx

(6)

one can extract the so-called N factor by taking the maximum valueof the amplitude exponent,

Nx max!

Z

x

x0

ix; ! dx

(7)

and then compare this value with a criticalN factor, which yields thetransition location.While this formulation has been applied in steady

Fig. 13 Wobble correction. By correlating a distinctive area of the reection line, the particle image can moved to its correct phase position.

Table 1 Uncertainty estimation of the PIV velocity measurement in the boundary layer [20]

Parameter Value Corresponding random error Conversion

Particle diameter 3 pixel 0.03 pixels Particle density 3 particles per 12 12 pixel2 0.03 pixels Particle displacement 10 pixels 0.02 pixels Gradient of the particle displacement 2 pixels=100 pixels 0.04 pixels Background noise 1/28 0.02 pixels Sum of random error/maximum expected error 0.14 pixels 0:126 m=sRoot of square sum of random error/uncertainty 0.065 pixels 0:058 m=s

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ow cases for many years, a new unsteady integration scheme for Nfactors was introduced by Radespiel et al. [6] by using the temporalevolution of amplication rates over time and space to computesuitable N factors along the airfoil surface for unsteady owproblems. Details of the formulation and its numerical imple-mentation are given in [6,26,27]. For low reduced frequencies, thesteady transition prediction is sufciently precise [26] and will beapplied in the present contribution.

V. Results

A. Verication of Numerical Simulations

A two-dimensional motion case (without rolling motion) for theoscillating SG04 airfoil with a reduced frequency of k 0:2 wasselected for the verication according to data from Pennycuick [17]and Herzog [18]. Hence, the interesting aspect of aerodynamicunsteadiness of the apping motion is included in this case. Table 2displays the detailed parameters. The ow case is typical for cruiseight of a bird in that amoderate amount of net thrust (cT 0:008) isproduced [7]. Net thrust is the sum of thrust and drag, consideringthat these individual forces can only be separated articially (seeWindte and Radespiel [26]).Both numerical simulations using the URANS code FLOWer, and

PIV measurements of the boundary layer for this two-dimensionalmotion case were published in Bansmer et al. [7]. Grid convergenceof the numerical solution was demonstrated using a mesh sequenceof a range between 136 28 and 1088 224. To quantify thecomputational uncertainty of the FLOWer results, a grid-convergence study was performed (see Fig. 14). It was found thatthe FLOWer results on a 544 112 mesh were essentially gridconverged. It was further shown that the numerical results of theFLOWer code are in good agreement with the experimental data. TheN factor sensitivity is depicted in Fig. 15. As expected, the plots withhigher critical N factors have their transition location furtherdownstream: the higher the criticalN factor, the lower the turbulencelevel, the more time disturbance waves need to grow, resulting in anincreased laminar ow length [7].The initial objective of this verication is to reproduce the previous

numerical results of the FLOWer code with the URANS code TAUused in this contribution. In contrast to FLOWer, TAU enables athree-dimensional transition prediction, which is used later for thethree-dimensional test case. To exclude discretization errors due todifferent mesh topologies, the original FLOWer mesh containing544 112 cells was also used in the TAU computations (see alsoFig. 16). For the computations, the freestreamMach number was setto 0.02. low-speed preconditioning was performed. A dual-time-stepping scheme was used for the time-accurate computations. Atemporal resolution of 500 physical time steps per period wasdened. For transition prediction, the critical N factor was set to 8.Although the ow quality at the test Reynolds number of 100,000 inthe wind tunnel, which was used for the validation experiments,demands for a choice ofNcrit 10 [6], we usedNcrit 8 to guaranteea robust run of the transition prediction. Later in the discussion,Ncrit 10 is considered again.When analyzing the rst results, the different convergence

performance between the two solvers was noticed. Figure 17 (left)demonstrates this difference by plotting the lift coefcient over onephysical time step. While the FLOWer code quickly reaches aconverged state with 70 inner iterations, the TAU code needs about

300 inner iterations. This is mainly due to more efcient algorithmsfor ux computations of the FLOWer code where the structuredmeshes in FLOWer enable a direct addressability of cell neighbors.TAU with its unstructured cell discretization compensates thisdisadvantage in computational time by using its parallelizationability.In the right part of Fig. 17, the distribution of lift and drag

coefcients and the transition location of the upper side of the airfoilare plotted against the nondimensional time, starting with t=T 0 atthe top dead center of the apping motion. The results of FLOWerand TAU are in good agreement. The computational uncertainty ofthe TAU results can therefore be assumed to be in the same order ofmagnitude as the FLOWer results. Minor discrepancies can beattributed to the following reasons:1) There are different calculations of the gradients for the

reconstruction of viscous uxes. TAU uses a GreenGauss ansatz forthis purpose, FLOWer deploys a difference scheme based on acurvilinear coordinate system due to its structured-mesh ability.2) For differences in the numerical dissipation, see the dissipation

stencils in Fig. 18. Whereas the neighbor cells i; j 1 and i; j 2

0 0.25 0.5 0.75 10

0.2

0.4

0.6

0.8

1

1.2

-0.2

-0.1

0

0.1

0

2

4

6

8

L1L2L3

Fig. 14 Convergence of lift, drag and transition location for differentgrid levels; L1 is the nest level with 1088 224 cells, apping motionwith k 0:2, Re 100; 000 [7].

0 0.25 0.5 0.75 10

0.2

0.4

0.6

0.8

1

0

2

4

6

8

N =6N =8N =10

Fig. 15 Transition prediction for different critical N factors; apping

airfoil with k 0:2 [7].

x/c

z/c

0 0.2 0.4 0.6 0.8 1

0

0.1

Fig. 16 Computational mesh of the SG04 airfoil with reduced

resolution of 272 56 cells.

Table 2 Motion parameters for the case of the oscillating airfoil

Parameter Value

Reynolds number Re 100; 000Reduced frequency k 0:2Chord length c 0:2 mGeometric mean angle of attack 0 4 degAmplitude of effective angle of attack eff 4 degPlunging amplitude z^ 0:1 mPitch/plunge phase difference 90 deg

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contribute to the dissipation in FLOWer, the dissipation in TAU isadditionally inuenced by the diagonal neighbors.In the next step, the three-dimensional simulation with TAU is

considered. To simulate the oscillating-wing segment with an aspect

ratio of 2, the fully structured mesh was extruded in the spanwisedirection with 32 cells. According to the results in Fig. 19, thisspanwise resolution is sufcient to capture the three-dimensionalaerodynamic effects on the wing segment. The mesh size of twomillion cells necessitated parallel computations using 128 CPUs. Asymmetry boundary was dened at both sides of the extruded mesh.The kinematics of the oscillating-wing segment were identical to thetwo-dimensional motion case with a reduced frequency of k 0:2.Again, a dual-time-stepping scheme was used for the time-accuratecomputations. A temporal resolution of 500 physical time steps perperiod was dened. The transition prediction was based on dataextracted from three streamlines that were equidistantly distributedin the spanwise direction. Along these streamlines, the eN methodwas applied using a critical N factor of 8.Figure 19 (left) compares the result of the two- and three-

dimensional TAU-simulation. The distribution of lift and dragcoefcients and the transition location of the upper side of the airfoilare plotted over one apping period. The agreement is very good, thefull three-dimensional simulation accurately reproduces the knowntwo-dimensional result. Several illustrations support the two-dimensional characteristics of the ow. For instance, Fig. 19 (right)shows a plot of the distribution of the specic dissipation rate on anisosurface of constant eddy viscosity for t=T 0:25. Similar to theaforementioned two-dimensional results, the transition is locatednear to the leading edge during the middle of the downstroke.Beyond, there are no spanwise variations of the transition location.Furthermore, the distribution of the shear stress on the surface of thewing segment demonstrate the two-dimensionality of the ow (seeFig. 20). The negative values of wall shear stress indicate the reverseow due to a laminar separation bubble, whose dimension does notvary in the spanwise direction. A qualitatively overlaid pressuredistributionwith its distinct pressure jump conrms the presence of aseparation bubble.

0 0.25 0.5 0.75 1

0.4

0.6

0.8

1

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0

0.2

0.4

0.6

0.8

1

TAUFLOWer

100 101 102

0.2

0.4

0.6

0.8

1

1.2

1.4

TAUFLOWer

Fig. 17 Comparison between FLOWer and TAU for the two-dimensional verication case, k 0:2, Re 100; 000. Left: Convergence of the liftcoefcient during one physical time step. Right: TAU and FLOWer give nearly the same result for one apping period.

Fig. 18 Dissipation stencils of FLOWer and TAU.

Fig. 19 Left: Comparison between two- and three-dimensional TAU computation for the oscillating-wing segment, k 0:2,Re 100; 000. Right: Theplot of the specic dissipation rate (turbulence modeling) on an isosurface of constant eddy viscosity shows a two-dimensional ow structure.

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B. Oscillating-Wing Segment for k 0:25A three-dimensional motion case (with roll motion, again see

Fig. 7) for the oscillating SG04 wing segment with an aspect ratio oftwo and a reduced frequency of k 0:25 was selected for thevalidation according to data from Pennycuick [17] and Herzog [18].The detailed motion parameters are presented in Table 3. Because ofthe spanwise variation of the plunging amplitude, the effective angleof attack is not constant along thewing. One feature of this motion isthe zero amplitude of the effective angle of attack at the left side of thewing segment. It is of interest to see, how lift and drag coefcients aredistributed at this location.A dual-time-stepping schemewas again used for the time-accurate

computations on the mesh with 544 112 32 cells. A temporalresolution of 500 physical time steps per period was dened. Bothside boundaries of the extruded mesh were dened as slip walls.Hence, the effect of the wind-tunnel sidewall boundary layers wasneglected.Moreover, the entire gridmoved alongwith the rigid-wingsegment, thereby neglecting the small effects of the upper and lowerwind-tunnel walls as well. For the computations, the freestreamMach number was set to 0.02 and low-speed preconditioning wasemployed. The transition prediction was based on data extractedfrom three streamlines that were equidistantly distributed in thespanwise direction. Along these streamlines, the eN method wasapplied using a critical N factor of 8. To study the inuence of thecritical N factor, additional computations were performed withNcrit 10. The transition prediction also covered the detection ofcrossow instabilities. However, the results of the linear stabilitysolver LILO demonstrated the absence of such instabilities.The distribution of lift and drag coefcients for one apping

period is plotted in Fig. 21. The top dead center of the motion is att=T 0. During the downstroke, t=T varies between 0 and 0.5. Theplot highlightsmany similarities comparedwith the two-dimensionalbehavior. For instance, the maximum lift (t=T 0:35) lags themaximum effective angle of attack (t=T 0:25) with a phasedifference of 35 deg. This is due to the dynamics of start-and-stopvortices in thewake of thewing segment and can be at least for a two-dimensional consideration explained with the inviscid theory ofTheodorsen [28]. The drag coefcient plot indicates negative dragduring the downstroke of the airfoil motion, i.e., the thrust is

produced in this part of the cycle mainly due to the KnollerBetzeffect: The effective angle of attack is large in the downstroke due toplunge velocity and creates lift; however, the negative geometricangle of attack generates a component of the lift in the upstreamdirection that is acting as thrust. The inuence of the criticalN factoron lift and drag is very limited and can bemerely observed during thewing upstroke. The below mentioned transition behavior explainsthisnding: the transition locations for both criticalN factors of 8 and10 are deviating from each other by 10% only.Caused by the rolling motion of the wing segment, the effective

angle of attack is not constant along the wing. Thus, each section ofthe wing segment has a different contribution to lift and drag (seeFig. 22). The plot shows the spanwise distribution of lift and dragcoefcients over one apping period. The amplitude of the effectiveangle of attack eff increases in positive y direction, the middlesection of the wing segment is at y=b 0:5. On the side with largeamplitudes of the effective angle of attack (y=b > 0:5), the largest liftvariations occur. In particular, the phase difference between effectiveangle of attack and lift obtains its smallest value in this region. Thisphase difference increases for decreasing eff because the pitch/plunge amplitude ratio L is increasing from 0.59 to 1 in this specialvalidation case. Themean angle of attack with its value of 4 deg doesnot change in the spanwise direction. Hence, the mean lift coefcientremains almost constant along the wing segment, cz 0:85. Thedrag coefcient also has its largest temporal variation for largeeff .However, the phase difference between effective angle of attack anddrag coefcient is nearly constant in the spanwise direction.The rolling motion of the wing segment also affects the transition

characteristics over one apping period (see Fig. 23). The distri-bution of the wall shear stress cf for t=T 0:375 and the overlaidpressure distribution indicate a laminar separation bubble, whosespatial extension is varying in the spanwise direction. However,nonexisting crossow instabilities and the fact that the tips of thewing segment are modeled as slip walls yield a nearly lineartransition distribution in the spanwise direction. Turbulence andtransition are further investigated and compared with experimentaldata for three spanwise sections (for eff f4:2; 2:7; 1:2g) inFig. 24. In each of the three sections, the transition moves toward theleading edge during the downstroke and back toward the trailingedge during the upstroke motion. The smaller the amplitude of theeffective angle of attackeff , the less the transitionmoves along theairfoil surface. The transition locations computed by the TAU codefor the two different criticalN factors show only small differences ofabout 10% and match well with the experimental determinedtransition locations from the PIV measurements.The approach to extract transition locations from PIV data shall be

discussed briey (see Fig. 25). Shown are the turbulent shear-stressdistributions xz from the PIV experiments and the numericalcomputations for three spanwise sections (efff4:2 deg; 2:7 deg; 1:2 degg) at four different phase angles starting

Fig. 20 Distribution of the shear stress on the oscillating SG04 wingsegment at t=T 0:25, k 0:2, Re 100; 000, from TAU computation.

Table 3 Motion parameters for the case of the three-dimensional

oscillating-wing segment

Parameter Left side Right side

Reynolds number Re 100,000 Reduced frequency k 0.25 Chord lengthc

0.2 m

Mean angle of attack 0 4 deg 4 degAmplitude of geometric angle of attack ^ 7.46 deg 7.46 degAmplitude of effective angle of attack eff 0 deg 5.22 degPlunge amplitude z^a 0.052 m 0.09 mPitch/plunge amplitude ratio L 1 0.59Phase difference plunge/pitch 90 deg

aThe difference of the plunging amplitude between the left and right sides of the wingcauses a roll motion.

0.25 0.5 0.75

0.8

1

-0.2

0

0.2Ncrit=8Ncrit=10

Fig. 21 Lift and drag coefcient over one apping period for the three-dimensional validation case, calculated with two different critical N

factors, k 0:25, Re 100; 000.

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from t=T 0 at the top dead center. The turbulent shear stresscharacterizes the viscous momentum transport across the boundarylayer and can be used to detect regions of turbulent ow. Thus, thetransition location in the measurements can be dened as thebeginning of the turbulent wedge that starts from the shear layer ofthe LSB. To avoid errors in its localization due to insufcientresolution of apping of the laminar part of the LSB, the point wherethe normalized Reynolds shear stress u0w0=U21 reaches 0.1% anddemonstrates a clearly visible rise is dened as the transitionposition. Derived from the Boussinesq approximation, the relation

u0w0 t%

@ u

@z @ w

@x

(8)

was used to recover the turbulent shear stress from the URANSsolution. The results demonstrate good agreement betweennumerical and experimental data. However, a closer look at theresults of the numerical simulation reveals strong local overshoots ofu0w0 at the location of laminar turbulent transition, whereas theexperimental u0w0 distribution is smooth along the surface. This isdue to the two-layer k-" turbulence model, whose dominant propertyin turbulence production is well known [29]. For futurecomputations, it will be interesting to see the performance ofReynolds-stress turbulence models, particularly because they areable to reproduce the anisotropy of turbulence for this stronglynonequilibrium ow case. Previous investigations of a two-dimensionalapping airfoil [7] revealed the relaminarization processfor 0:5 t=T 0:75 to be the most difcult to simulate.Interestingly, the relaminarization process in the present three-dimensional simulation is well captured.Can the three-dimensional aerodynamics of the apping-wing

segment simply be reproduced as a spanwise sequence of two-dimensional airfoil aerodynamics at equivalent effective angle ofattack? Figure 26 answers this question. Shown are lift and dragcoefcients and transition location over one apping period for twospanwise sections of the wing segment. The equivalent two-dimensional aerodynamics of the SG04 airfoil, which werecomputed with TAU, are overlaid, assuming the airfoil moves withthe localmotion of eachwing section.Hence, the samedistribution ofthe effective angle of attack over one apping period is ensured foreach two-dimensional apping case with its corresponding wing-section counterpart.Compared with the two-dimensional case, there is less lift

produced for the wing section with a large amplitude of the effectiveangle of attack (left plot) during the apping downstroke in the three-dimensional case. This is due to the vortex dynamics in the wake of

Fig. 22 Spanwise distribution of lift and drag coefcients over one apping period, Ncrit 10, k 0:25, Re 100; 000.

0.0090.0070.0040.0020.000

-0.002-0.004-0.006-0.008-0.010

Fig. 23 Shear-stress distribution on the wing segment for t=T 0:375,k 0:25, Re 100; 000, from TAU computation. The negative values ofshear stress and the overlaid pressure distribution indicate the presenceof a laminar separation bubble, whose spatial extension varies in the

spanwise direction.

1

2

3

4

5

6

7

0 0.25 0.5 0.75 10

0.2

0.4

0.6

0.8

1

2

3

4

5

6

0 0.25 0.5 0.75 1

0

2

4

6

8

TAU, N =8TAU, Ncrit

crit=10

experimenteff

0 0.25 0.5 0.75 10

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

Fig. 24 Comparison of the transition locations over one apping period with experimental PIV data for three selected spanwise sections.

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Fig. 25 PIV data and numerical simulation.

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thewing segment. The unsteady part of the 3-Dwake is composed ofstart-and-stop vortices with a strength that varies in the spanwisedirection, and in between these spanwise structures the formation of astreamwise vortex street takes place, so that conservation of vorticityis fullled. The induced velocities of the streamwise vortex streetchange lift and drag coefcients of the wing segment as shown inFig. 26. In particular, they explain the induced drag of the wingsegment that lowers the propulsive efciency p of the appingmotion. According to Windte and Radespiel [26], the propulsiveefciency can be determined by

ct cx cx;stat0 (9)

p RT0 ctU1 dtR

T0 cx _x cy _y cz _z cm _c dt

y=bconst

(10)

This equation also contains the thrust coefcient ct. In contrast tothe thrust determination of conventional aircraft, thrust and drag canonly be separated articially for apping yers. In Eq. (9), this isdone by subtracting the drag of the SG04 airfoil at the mean angle ofattack from the force coefcient cx. Figure 27 shows the spanwisedistribution of the propulsive efciency for the three-dimensionalTAU computation. The efciency is not constant along the span. Asexpected, on the side with large amplitudes of the effective angle ofattack eff and low pitch/plunge amplitude ratios L (y=b > 0:5)the propulsive efciency is larger. The efciencies of the two-dimensional reference cases of Fig. 26 are indicated as diamonds andprove the above mentioned hypothesis: the formation of streamwise

vortices in the three-dimensional case creates an efciency loss ofabout 10%.

VI. Conclusions

A birdlike wing segment is considered, oscillating with anominally three-dimensional combined pitchplungeroll motion ata reduced frequency of k 0:25 in the transitional Reynolds numberrange of 100,000. A sophisticated numerical computation schemebased on the unsteady Reynolds-averaged NavierStokes solverTAUwas set up. To validate the numerical simulations, the boundarylayer of the oscillating-wing segment was investigated with high-resolution stereoscopic particle image velocimetry. Good agreementin the ow prediction has been demonstrated. The transitionlocations of experimental and numerical result are especially in goodagreement. The two-layer k-" turbulence model delivers generallyadequate results; however, systematic discrepancies betweensimulation and experiment were found regarding the turbulenceintensity. The results demonstrate that the numerical simulationscheme provided in this contribution may be deemed as acceptabletools for conceptualization and parametric studies to understand theapping-wing aerodynamics of natural yers. An analysis of thethree-dimensional simulation also highlighted its signicance: thethree-dimensional aerodynamics of the apping-wing segmentcannot be reproduced as a spanwise sequence of two-dimensionalresults due to the complex vortex structure in the wake.

Acknowledgments

As a part of the priority program SPP 1207, this project is fundedby the German Research Foundation (DFG). The majority ofcomputations were performed at the facilities of the North-GermanSupercomputing Alliance (HLRN). The authors thank A. Probst, S.Mahmood, and N. Krimmelbein for valuable discussions and theirsupport.

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[32] Ellenrieder, K., Parker, L., and Soria, J., Flow Structures Behind aHeaving and Pitching Finite-Span Wing, Journal of Fluid Mechanics,Vol. 490, 2003, pp. 129138.doi:10.1017/S0022112003005408

[33] Parker, L., Soria, J., and Ellenrieder, K., Thrust Measurements from aFinite-Span Flapping Wing, AIAA Journal, Vol. 45, No. 1, 2007,pp. 5870.doi:10.2514/1.18217

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F. LadeindeAssociate Editor

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