773.1

ICAS 2000 CONGRESS

Abstract

This paper describes the application of thenoncommercial CFD-code FLOWer to theproblem of a sinusoidally pitching NACA 0012airfoil with high amplitude and reducedfrequency under incompressible flow conditions.As FLOWer allows the approximate solution ofthe nonlinear conservation laws governingviscous fluid flow, i. e. the Navier-Stokesequations, a numerical investigation of theunsteady boundary layer separation occurringduring such a motion becomes feasible.Employing FLOWer under the incompressibleconditions that correspond to the regardedexperimental regime required an extensivepreliminary analysis of the numericalparameters imbedded into the code. Havingbeen primarily developed for transonic flow, itwas possible to adjust FLOWer to theincompressible flow problem after studying thebehavior of its various mechanisms foraccelerating convergence and increasingstability. The necessary methodology to obtainoptimal parameter settings was developed bycritically examining steady state solutions atlow angles of attack. The knowledge gainedfrom these cases was then applied to simulatethe sinusoidal movement of the NACA 0012airfoil. FLOWer’s accuracy in predicting thephenomena of the unsteady boundary layerseparation was then assessed.

1 Introduction

In the field of unsteady aerodynamics the termdynamic stall is frequently used to describe thecomplex fluid mechanical phenomena that occur

* Director: o. Prof. Dr.-Ing. Boris Laschka

during extreme incidental movements of anairfoil beyond the angle of static stall. Theseprocesses are characterized by unsteadyboundary layer separation, where a multitude ofdynamic and viscous factors influence the flowfield evolution. Extreme variations to the steadystate flow separation become apparent. Initialinvestigations on this subject were performed inthe sixties after stall induced rotor flutter hadbeen observed during helicopter development.While in forward flight retreating rotor bladesexperience a combination of high incidencesand low relative velocities of the oncomingfreestream, resulting in an unsteady separatedflow field in regions above the upper bladesurface. With these dynamic phenomena havingbeen localized as the cause for the flutterproblem, further studies became necessary, as ameans to extend the rotor’s performance beyondthe stall determined barrier.

A significant research breakthrough wasachieved by confirming the existence of adistinct dynamic vortex structure on the suctionside of the rotor blade. The use of an airfoilpitching in incidence allowed the reproductionof these physical effects under laboratoryconditions for experimental analysis. Numerousmeasurement campaigns have been performedover the past years, in order to obtain andexamine the dynamic stall evolution for avariety of wing and airfoil geometries, incidencemotions and freestream conditions. Results arepredominantly available for infinite wings basedon the classic symmetrical rotor airfoils NACA0015 and NACA 0012, with a sinusoidalincidence motion being imposed to simulate thecyclic blade pitch. Significant publications onthis matter include the findings of Raffel,Kompenhans and Wernert [8] [10], pertaining to

NUMERICAL SIMULATIONOF A PITCHING NACA 0012 AIRFOIL

Dipl.-Ing. Alexander PechloffLehrstuhl für Fluidmechanik*, Technische Universität München

Keywords: numerical simulation, Navier-Stokes equations, dynamic stall

Alexander Pechloff

773.2

a recent particle-image velocimetry campaign ofthe two-dimensional unsteady boundary layerseparation above an oscillating airfoil. In ac-cordance with [8] and [10] four sequentialphases of the dynamic stall process can bediscerned:

The initial phase is characterized bycompletely attached flow during the airfoil’supstroke motion, persisting even beyond theangle of static stall. As a distinguishing feature,a thin layer of reverse flow in proximity to theairfoil’s surface becomes evident. At firstoccurring near the trailing edge, the layerquickly progresses towards the leading edgewith increasing angle of attack, where iteventually initiates the so called dynamic stallvortex.

Development and progression of thisleading edge vortex defines the second and mostsignificant phase of the flow oscillation. Withthe deceleration of the reverse flow layer at theleading edge, unsteady boundary layerseparation commences and the dynamic stallvortex is generated. As the incidence motioncontinues, the vortex increases in diameter andintensifies, subsequently moving downstreamover the airfoil’s surface. Because the vortex isin direct contact with the airfoil surface, thehigh circulation velocities result in a verypronounced, yet spatially limited zone of lowpressure. This occurrence is responsible formaintaining high lift even after the primarysuction peak at the leading edge collapses. Ingeneral maximum lift coefficients obtainedduring dynamic stall are significantly higherthan their steady counterparts.

The previous phase concludes with theonset of post stall vortex shedding. Assecondary vortex structures evolve near theleading and trailing edge under completelyseparated flow, the dynamic stall vortexdetaches from the airfoil’s surface andprogresses into the wake. Consequently, thesecondary suction peak induced by the dynamicstall vortex is no longer sustained, resulting inthe collapse of the lift coefficient. Theseprocesses, which are associated with the down-stroke motion of the airfoil, display a highdegree of aperiodicity. Hence, a cycle to cycle

reproduction and acquisition of the phenomenais made difficult.

In the final phase of dynamic stallcontinuous flow reattachment occurs, beginningat the leading edge. The area of unseparatedflow is repressed, with the remaining vortexstructures dissolving into the wake. Reattach-ment is completed on the trailing edge towardsthe end of the downstroke.

At the Lehrstuhl für Fluidmechanik of theTechnische Universität München research onthe subject of dynamic stall has also beenconducted, with Ranke [9] giving a universaldefinition for the two-dimensional unsteadyboundary layer separation. The theoriespresented in [9] were applied to numericalmethods and verified by Ranke throughexperiments performed on a sinusoidallypitching NACA 0012 airfoil with a reducedfrequency of kred = 0.3 at Ma∞ = 0.1 and Re∞ =2.5 ⋅105. The acquisition of this test case pro-vided valuable reference data for future studies.

More recently research to extended thisknowledge about the unsteady boundary layerseparation to the third dimension has beenunderway. Unprecedented measurements toinvestigate the dynamic stall on an infiniteswept wing with the above airfoil specificationsfor various reduced frequencies are to besupplemented by a database of numerical resultsfor comparative studies. In the first phase aunsteady Navier-Stokes code is employed tosimulate two-dimensional test cases andevaluate the accuracy of the numericallyobtained flow fields. Unlike the coupling ofunsteady boundary layer methods with inviscidouter flow solutions the use of a Navier-Stokessolver allows the computation of the flow fieldbeyond the onset of the leading edge vortex,rendering the complete dynamic stall processand facilitating the analysis.

In the following chapters the numericalbasics of the Navier-Stokes solver will bepresented and its application to the dynamicstall scenario will be discussed. An extensiveanalysis of the numerical parameters allowedthe adaptation of the code to the incompressibleflow regime of the experimental test case, with

773.3

NUMERICAL SIMULATION OF A PITCHING NACA 0012 AIRFOIL

the evaluation of the simulation results alsobeing given herein.

2 Basics of the Numerics

The numerical simulation of the sinusoidallypitching NACA 0012 airfoil is realized with theflow solver FLOWer, a noncommercial codedeveloped by the German aerospace center(DLR), Daimler Chrysler Aerospace (DASA)and selected German universities for transonicflow around transport type aircraft. It is basedon a finite volume discretization of theReynolds-averaged Navier-Stokes equations,which are solved either in an explicit or implicitformulation using a multistage Runge-Kuttaintegration technique.

In general the explicit scheme is used toobtain initial steady state solutions beforeswitching to the implicit scheme, allowing timeaccurate calculations for the unsteady case. Thismethod, which is also known as dual time-stepping, circumvents the stability limitationsimposed by the time increment of the explicitscheme.

A solution for an unsteady flow at adistinct moment in physical time results fromthe iterative determination of a pseudo steadystate for that instant. The necessary pseudo timeintegration, however, allows the use of allstability enhancing and convergence accel-erating methods of the explicit scheme, withoutsacrificing time accuracy. In this respectFLOWer offers such features as local timestepping, multigrid cycles, implicit residualaveraging and adjustable artificial dissipation.The last feature ensues from the employednondissipative finite volume discretization,which necessitates the introduction of artificialviscous terms for maintaining stability. Theimplemented Baldwin-Lomax turbulence modelwill be used in the following to simulate randomflow fluctuations. A complete description ofFLOWer’s properties is given in [3], withfundamental algorithms having been derived byKroll [4] and Radespiel [7].

2.1 Mathematical Model, Discretization andStability

The Reynolds-averaged Navier-Stokes equa-tions, governing the statistically treated un-steady viscous flow, can be written in conser-vative integral form as:

T

V V

vc

)E,w,v,u,(W

;0dSn)FF(dVWt

ρρρρρ=

=−+∂∂ ∫∫∫ ∫∫

∂&

&

&

(1)

where W&

denotes the solution vector containingthe five unknown conservative variables:density ρ, the three Cartesian components ofmomentum ρu, ρv, ρw and the total energy ρE.V defines an arbitrary control volume, which islimited by its surface ∂V, with n

& representing

the normal vector of the control surface. cF and

vF represent the tensors for the convective andviscous flux, respectively:

,

)()()(

)(

)(

)(

2

2

2

++++++++

++++++++

=

=

zyx

x

zyx

zyx

zyx

ewpwEevpvEeupuE

epwewvewu

evwepvevu

euweuvepu

eweveu

cF

zy&&&

&&&

&&&

&&&

&&&

ρρρρρρ

ρρρρρρρρρ

(2)

++++++++

=

zyx

zzzzyxzx

zyzyyyxyx

zxzyxyxxx

eee

eee

eee

0

vF

y

ΠΠΠσσσσσσσσσ

&&&

&&&

&&&

&&&

(3)

with

.)(

)(

)(

zzzzzyzxz

yyyzyyyxy

xxxzxyxxx

eqwvu

eqwvu

eqwvu

&

&

&

&

&

&

−++=Π

−++=Π

−++=Π

σσσ

σσσ

σσσ

(4)

The vectors ��

e x , �

��

e y and ��

��

e z define the three

Cartesian coordinate directions within the above

Alexander Pechloff

773.4

tensors. Pressure p is calculated with the ther-mal equation of state given as

.))(2

1()1( 222 wvuEp ++−−= ρκ (5)

In equation (5) κ resembles the ratio of specificheats.

All components of the shear stress tensorσare governed by the material laws for New-tonian fluids under consideration of the Stokeshypothesis, while the components of the heatflux qx, qy and qz correspond to Fourier’s law ofheat conduction. Molecular viscosity is calcu-lated according to Sutherland, whereas the heatconductivity results from the molecular vis-cosity µL and the Prandtl number. In the case ofturbulent flow the modeled eddy viscosity µT isadded to the molecular viscosity, just as theturbulent heat conductivity is added to thelaminar one. For the sake of brevity thesewidely known equations will not be presentedhere, as they are detailed in [3]. This referencealso includes information about the usedalgebraic turbulence model.

By employing a finite volume discre-tization based on central averaging to theNavier-Stokes equation (1) a system of ordinarydifferential equations is obtained with

,)(1

,,,,,,,,

,, kjiv

kjic

kjikji

kji DRRV

Wdt

d &&&&

−−−= (6)

allowing the calculation of the solution vector ateach node of a structured grid. The right handside of (6) contains the approximation of theconvective and viscous fluxes (

��

��

R i , j, kc ,

��

��

R i , j, kv )

representing the net flux of mass, momentumand energy for a particular control volume Vi,j,k

arrangement surrounding the grid node (i,j,k).��

D i, j ,kc is the additional artificial term introduced

into the equation, in order to dampen numericaloscillations induced by the solution process inthe otherwise nondissipative scheme. Theoperator itself is comprised of a blend of secondand fourth differences of the flow variablesexercised within the semi-discrete system, in

order to preserve the conservation form. Thistechnique was initially developed by Jameson[2]. A strict mathematical explanation of theoperators mode of functioning has not beengiven. However, numerical experiments haveshown that the method is very capable inproducing stable solutions, especially in respectto viscous flow as Martinelli [5] describes.

Within FLOWer the user can exert controlover the artificial viscous terms by adjusting theusually small parameter values of k(2) and k(4).While the parameter k(2) controls the first orderdissipation introduced locally in shock vicinity,k(4) sets the more general third order backgrounddissipation applied to the global flow withoutdiscontinuities. For the incompressible flowregime, where no shocks are to be expected, theinfluence of the parameter k(4) on the numericalsolution has to be analyzed, whereas k(2) canremain at its default setting of 1/2. Especiallythe trade off between stability and minimalsolution distortion, resulting from too muchartificial viscosity, has to be evaluated.

Besides adjusting the numerical dissipationthe user retains the ability to control stabilityand convergence of the flow solution byadjusting the residual averaging algorithmnimplemented in FLOWer. For the three-dimensional computational domain the implicitand discretized smoothing equation is given as

,

)1()1()1(

)(,,

)(,,

mkji

mkji

P

P&

&

=

∆∇−∆∇−∆∇− ζζζηηηξξξ εεε(7)

where )(,,

mkjiP

�

denotes the m-th stage residual

vector of the multistage Runge-Kutta scheme, asapplied to the ordinary differential equation (6).Within the factorized averaging equation (7)∇ and ∆ represent the normal forward andbackward difference operators, respectively,whereas εξ, εη and εζ are the positive smoothingcoefficients in the individual coordinatedirections. In studies presented here only εxyz =εξ = εη = εζ will be examined. The averaged

residual vector )(,,

mkjiP

�

results from the iterative

solution of equation (7) at distinct Runge-Kutta

773.5

NUMERICAL SIMULATION OF A PITCHING NACA 0012 AIRFOIL

stages. By employing this technique theallowable time increment of the integrationprocess can be extended far beyond the actualstability limit. The user has two possibilities toadjust this feature to the individual flowproblem, by setting either constant or adaptivesmoothing coefficients.

The first method presents a simple way ofdampening numerical instabilities. According toMartinelli [5] optimizing the smoothingcoefficients for a certain test case requires acertain degree of experience, in order to obtainthe desired convergence behavior. However, therequired numerical investigations for deter-mining these empirical coefficients can becomequite extensive.

In contrast the adaptive form of implicitresidual averaging uses smoothing coefficientsthat are a function of the local characteristicwave speeds in the individual cell. The userexerts control on the algorithmn by defining anintervall for the lower and upper value of theallowed smoothing coefficient. For this methodthe user once again has to obtain knowledgeabout the selected test case’s stability behavior,somewhat limiting the adaptive techniquesadvantages. Regardless of the chosen techniquethe optimization of the implicit residualaveraging is prerequisite to obtaining stableNavier-Stokes solutions. Especially the cellvolumes of high aspect ratio, as employed in theresolution of boundary layer gradients, limit theallowed time increment and define the stabilityregion of the scheme.

The discretization of the physical domainsurrounding the airfoil is accomplished with atwo-dimensional single block structured gridbased upon a ‘C-topology’ as depicted in Fig. 1.Elliptical smoothing of the grid was performed,in order to satisfy FLOWer’s requirement ofsecond-order accurate spatial discretization. Theairfoil’s contour is digitized with 257 individualpoints. In preliminary investigations the gridwas composed of 384 × 64 cells, with the firstgrid line’s off-body distance being 0.9⋅10-3

chord lengths. Numerical studies revealed,however, that this discretization was insufficientin resolving the boundary layer properly. Hence,

the number of cells was increased to 384 × 96,while the off-body distance was reduced to0.5⋅10-3 chord lengths. Fig. 2 depicts theresolution of the 384 × 96 grid in proximity tothe airfoil surface. The results presented in thispaper all pertain to this grid.

In the following a small excerpt of theextensive numerical studies performed withFLOWer on turbulent steady state cases withlow angles of attack (Ma∞ = 0.1, Re∞ = 2.5⋅105)will be given. The analysis of FLOWer’sbehavior regarding convergence accelerationtechniques such as local time stepping andmultigrid cycles can be reviewed in Pechloff[6]. The results for the implicit residualaveraging and artificial dissipation however arediscussed.

2.2 Evaluation of the Implicit ResidualAveraging

The initial motive for studying the implicitresidual averaging evolved during the transitionfrom the 384 × 64 grid to the finer 384 × 96grid, due to the integration scheme exhibitingextreme sensitivity to the higher resolution ofthe boundary layer. Previous calculations on thecoarse grid performed beyond FLOWer’stheoretical stability limit of CFL = 3.75(Courant-Friedrich-Levy number) [5] showedadequate convergence and stability withFLOWer’s default setting of the smoothingcoefficient εxyz = 0.7. Attaining a stable steadystate solution with the finer mesh, however, wasno longer possible with this setting, with a shiftof the integration algorithm’s region of stabilityhaving become evident. In order to determinethe new εxyz parameter values for residualaveraging with constant coefficients, thestability region in respect to different CFL-numbers (3.75, 5.625 and 7.5) was examined.As the user exerts control over the integrationalgorithmn’s time increment by setting the CFLparameter, it becomes necessary to adjust theresidual averaging accordingly.

Criterion for localizing the εxyz interval isthe convergence of the steady state solutionbelow a root mean squared (RMS) density

Alexander Pechloff

773.6

residual of 1.0⋅10-5, after which the calculationis terminated. Reaching a limit of 1500multigrid cycles is defined as the secondarycriterion. Beyond this limit a solution will beclassified as not convergent. Settings for themutigrid cycle were optimized in preliminaryinvestigations not presented here and remainedunmodified in the course of this evaluation. Itshould be noted that a ‘symmetrical W’multigrid cycle type had been found to be veryefficient for convergence acceleration.

Evaluating the implicit smoothing algo-rithmn’s behavior, as performed in Fig. 3, adistinct boundary of convergence for eachvariation of the CFL number becomes obviousand presents a specific interval for the setting ofthe respective smoothing coefficient εxyz.Furthermore within each stability region anoptimal value for the smoothing coefficientexists, accomplishing convergence down toengineering accuracy with a minimum amountof multigrid cycles. Regarding the optimalsmoothing coefficient for each individual CFL-number, a reduction in computational effort by9% can be realized when increasing the CFL-number from 3.75 to 7.5. The results of theevaluation are especially interesting for theadaptive form of implicit residual averaging,because it is now possible to properly set theupper and lower boundary of the smoothingcoefficient’s interval in respect to the CFL-number.

As the increase of the NACA 0012 airfoil’sangle of attack is in general associated with adeterioration in convergence behavior, it can beexpected that the convergence boundariesobtained for α = 0° will constrict, thus reducingthe allowed coefficient interval for the adaptivesmoothing algorithm. Consequently, withoutresetting the interval boundaries the amount ofmultigrid cycles required to breach the residualtolerance level would continuously increase, upto the point where a steady state solution can nolonger be attained. In regards to the unsteadysimulation of an incidence motion, such as willbe presented in later chapters, a perpetualmanual adjustment of the adaptive intervalwould become necessary.

A comparison of the constant coefficientsmoothing algorithmn at optimal settings (εxyz =1.9, CFL = 7.5) with the adaptive one limited tothe stability region favors the first method inrespect to convergence behavior (Fig. 4).Hence, under incompressible flow conditionsthe method of adaptive residual averaging isdeemed not as effective as the constantcoefficient technique once the optimal settinghas been retrieved. This result remains indiscrepancy with the effectiveness of theadaptive technique as described in Radespiel[7]. To which extent the residual averagingdistorts the solution was not examined in thisstudy.

2.3 Influence of the Artificial Viscosity

As was mentioned in chapter 2.1, the user isable to control the amount of backgroundviscosity artificially introduced into the solutionby setting the parameter k(4). However, this hasto be done very cautiously, because the stabilityenhancing feature of the artificial dissipationcomes with a trade off in regards to solutionaccuracy. In order to obtain an optimal settingfor the third order dissipative constant k(4) onehas to examine stability behavior and deviationof the solution from the physical flow.

The first study investigates the effect of ak(4) variation on the convergence behavior of thecalculation. Once again steady state solutions atα = 0° are examined, with the CFL-number andconstant smoothing coefficient set to CFL = 7.5and εxyz = 1.9, respectively, as a result of theprevious evaluation (chapter 2.2). In accor-dance, steady state calculations are terminatedeither after reaching a RMS density residualsmaller than 1.0⋅10-5 or exceeding 1500multigrid cycles. Variation of the dissipationconstant k(4) was performed by successivelyhalving the default value of 1/64 as a means ofdecreasing dissipation levels. On the other handan increase in background dissipation wasachieved by successively doubling this initialvalue. Results of this study are depicted in Fig.5. As a principal observation it can be statedthat the increase in artificial backgrounddissipation leads to an accelerated convergence

773.7

NUMERICAL SIMULATION OF A PITCHING NACA 0012 AIRFOIL

and vice versa. This behavior is a direct conse-quence of the stabilization quantity introducedinto the solution algorithmn. With a setting ofk(4) = 1/16 fastest convergence to the steadystate solution is achieved, requiring only 298multigrid cycles and exhibiting minimal oscil-lations in the density residual’s progression. Incontrast the reduction of k(4) to 1/512 requires1285 multigrid cycles to achieve the specifiedtolerance level, while showing greater oscil-lations in the residual.

The behavior of the calculation at the twoboundaries of the convergence region is alsoquite different. Further increase of k(4) from 1/16to 1/8 leads to a sudden convergence limit, withthe cause being identified as local instabilities ofthe algorithm occurring near the airfoil’sleading edge. Continued reduction of thebackground dissipation to a minimal value,however, leads to an asymptotic progression ofthe residual towards the specified accuracy.Therefore the lower convergence boundary fork(4) is determined by the global destabilizationof the solution, with termination occurring afterthe maximum amount of multigrid cycles. Eventhough rapid convergence can be achievedthrough a setting of k(4) = 1/16, the quality of thesolution has to be regarded as a second criterionbefore deciding on an optimal setting.

Because the introduction of the artificialviscous terms has a similar effect in dampeningnumerical oscillations and accelerating con-vergence as the implicit residual averaging, therelationship between these two methods has tobe examined first. This is accomplished bylocalizing the boundary of convergence inregards to the smoothing coefficient εxyz for eachvariation of the third order dissipation constantk(4). The methodology is equivalent to the onedescribed in chapter 2.2, but performed for a 2°airfoil incidence. As becomes evident in Fig. 6an increase of the third order dissipationconstant leads to a broadening of the allowablesmoothing interval. Hence, the parameter εxyz

doesn’t require an optimal setting as a means toachieve adequate convergence behavior, be-cause solution stability is increasingly suppliedby k(4). In the most extreme case of k(4) = 1/16 a

smoothing interval between 1.5 and 2.1 for εxyz

is still sufficient to obtain the specified residualtolerance. Complementary to this behavior acontinuous reduction of dissipation levels makesthe integration algorithmn more sensitivetowards the implicit residual averaging. As aconsequence either the employed smoothingcoefficient or the adaptive interval have to be setmeticulously as a means of stabilizing thecalculation.

Finally the aspect of solution distortion isanalyzed by observing the drag coefficient’svariation in relation to the applied stabilizationmethods. Results are depicted in Fig. 7. Implicitresidual averaging for a fixed level of back-ground dissipation portrays hardly any influenceon the airfoil’s drag coefficient Cd. On the otherhand variation in the third order dissipationconstant has significant impact on Cd, which isinherently coupled with the flow’s viscosity. Aminimal drag coefficient for k(4) = 1/32 withinthe interval of 1/16 and 1/512 becomes evident.Because of the artificial dissipative termsintroduced into the discretized equation (6), thesolution in general tends to have a higher degreeof viscosity than the actual physical flow.Hence, by localizing the minimum of the dragcoefficient it can be argued, that a setting of k(4)

= 1/32 achieves a solution with the leastdistortion, with Cd being closest to the actualphysical value. Under consideration of the sta-bilizing and convergence accelerating effects ofthe numerical background dissipation, a settingof k(4) = 1/32 is deemed optimal for approx-imating physically accurate solutions.

3 Unsteady Simulation

3.1 Method

To validate FLOWer’s ability to properlypredict the occurring unsteady flow phenomenaassociated with deep dynamic stall a sinusoidalpitching motion corresponding with the ex-perimental test case supplied by Ranke [9] (Ma∞= 0.1, Re∞ = 2.5⋅105) was conducted. The in-cidence law given by

)tf2(cos1010)t( ⋅⋅°−°= πα (8)

Alexander Pechloff

773.8

governs an airfoil motion that consist of a ± 10°incidence variation about a mean incidence of10°, with the pitching axis located at 25% chordlength from the leading edge. The values offrequency f = 2.0 Hz, freestream velocity U∞ =12.6 ms-1 and chord length c = 0.3 m lead to areduced frequency of

.30.0U

cf2kred ≈=

∞

π(9)

Employing FLOWer’s dual time steppingalgorithmn, a Navier-Stokes simulation of themotion defined by (8) can be accomplished aftersupplying an initial steady state solution. In thiscase a steady state solution of the oscillation’slower dead center was used, which was obtainedthrough the application of methods described inchapters 2.2 and 2.3. A single period ofoscillation (T) is discretized with 100 individualtime steps, each being the equivalent of ∆t/T =0.01 nondimensional phase increments. In orderto eliminate transient phenomena from thesolutions, oscillations beyond the initial cyclewill have to be completed.

Regarding the numerics, parameter settingsrespective to multigrid, local time stepping,implicit residual averaging and dissipation aretaken from the steady state solution andtransferred to the dual time stepping scheme.Because FLOWer calculates a pseudo steadystate for each physical time step, it was able toachieve satisfying results by using this method.However, the tolerance for the RMS densityresidual had to be raised to 1.2⋅10-5 over theentire oscillation, as a means of terminating thecalculation at each instance. During the phase ofdynamic stall vortex shedding convergence tothis specified accuracy was not attainable, dueto the highly unsteady nature of the occurringflow field and the size of the selected timeincrement. For these solutions the convergenceof the integral coefficients such as lift Cl, dragCd and moment Cm had to be observed, in orderto determine if the pseudo steady state had beenreached. The maximum amount of pseudo-iterations was limited to 150 multigrid cycles.

The results presented in the followingchapter relate to the 384 × 96 grid, with no griddeformation occurring over the incidencemotion, as FLOWer adjusts the boundaryconditions accordingly.

3.2 Description and Discussion of Results

To denote up- and downstroke the symbols ↑and ↓ are introduced, respectively. Labels con-tained in figures are referred to by {x}. Flowfields are visualized through the total velocityutot.

3.2.1 Evaluation of the Lift Coefficient Cl overthe Incidence Angle αAs becomes evident in Fig. 8, significanttransient phenomena is eliminated after sixcomplete cycles of the sinusoidal pitchingmotion. Even though good reproducibility of thelift coefficient can be obtained for the upwardstroke, strong aperiodic effects can be observedduring the downward motion, attributable to thevortex shedding occurring beyond 19.30°↓incidence (t/T = 0.56 {C}). However, it stillremains to be seen if the aperiodic effectsencountered during the downstroke areprimarily a chaotic phenomena or are the resultof a cycle dependent phase shift, as cycles twoand four would indicate. The ascending branchof the hysteresis is predominated by attachedflow, with the thin reverse flow layer firstappearing at 10.63°↑ (t/T = 0.26) and reachingthe leading edge at 19.30°↑ (t/T = 0.44 {B}). Atthis instance the dynamic stall vortex is induced,its evolution and intensification leading to amaximum lift coefficient of Cl max = 2.015during the downstroke (19.69°↓ , t/T = 0.54).

Within the next two time increments(19.69°↓ , t/T = 0.54) the lift collapses, as a newvortex evolves in proximity to the trailing edge,progressively separating the dynamic stallvortex from the airfoil surface. The subsequentreattachment of the dynamic stall vortex duringthe first half of the downward motion results ina secondary maximum lift coefficient of Cl max2

= 0.471 (13.09°↓ , t/T = 0.70), being preceded bya minimum lift coefficient of Cl min = -0.002(14.82°↓ , t/T = 0.67). The onset of flow

773.9

NUMERICAL SIMULATION OF A PITCHING NACA 0012 AIRFOIL

reattachment can be localized at an incidence of11.87°↓ (t/T = 0.72 {D}).

3.2.2 Evaluation of the Pressure CoefficientVariation Cp over the Incidence Angle α and ofSelected Flow FieldsFrom the Cp-α-x/c contours depicted in Fig. 9and 10 eight significant properties of thedynamic stall process can be extracted. Labelsused in this context refer exclusively to thesetwo figures. The upward motion of the airfoil,which is predominated by attached flow, ischaracterized by the development of a primarysuction peak above the leading edge {1} (Cpmin

= - 10.2, 18.76°↑ , t/T = 0.42). This results fromhigh flow velocities occurring as fluid passesaround the airfoil’s leading edge. After theappearance of the dynamic stall vortex {2}(19.30°↑ , t/T = 0.44, Fig. 11) and itsdevelopment (Fig. 12), which has a circulationcenter located horizontally at approximately25% chord length, the leading edge pressureformation collapses. The cause for this processis identified in the intensification and expansionof the dynamic stall vortex, detaching the flowaround the leading edge.

Behavior of the pressure coefficient duringthis upward motion corresponds well withresults from Coton [1], who performed so calledramp-up experiments with airfoils pertaining todynamic stall. Even though a linear incidencelaw with a constant maximum angle governedthe airfoil motion, a similar drop in the primarysuction peak was observed after the inception ofthe leading edge vortex. Coton having acquireddata for a nonperiodic flow, however, furthercomparisons of the Cp-α-x/c distributions arenot possible.

Returning to the sinusoidally pitching air-foil, the secondary suction bulge, which isinduced by the dynamic stall vortex, achieves itsminimum value of Cpmin2 = -5.2 {2} at an in-cidence of 19.82°↑ (t/T = 0.47). At this point thevortex has reached its maximum circulationintensity, which will steadily decrease as thevortex moves downstream and continues toexpand. While the primary suction peak reducesduring the downward motion {3}, the dynamicstall vortex as characterized by the secondary

suction bulge passes over the airfoil surfacetowards the trailing edge {4}. It becomesevident, that the secondary suction effect, whichis primarily responsible for maintaining highlift, decreases with the progression of thepitching motion. At the same instance, however,the vortex induced area of low pressure extendsfarther across the airfoil surface, consequentlyshifting the moment of maximum lift Cl max =2.015 into the downstroke (19.69°↓ , t/T = 0.54).In this respect Fig. 13 shows the dynamic stallvortex at an incidence of 19.30°↓ (t/T = 0.56),having reached its largest diameter and exten-ding over approximately 75% of the airfoil’schord length.

With the subsequent shedding of thedynamic stall vortex, the suction ridge {4}subsides and gives way to a new low pressurepeak forming at the trailing edge {5}. It isinduced by the evolution of the trailing edgevortex, which gains on influence during thedownward motion before drifting into the wake.As a result the trailing edge suction peakdiminishes {6}, coinciding with the reattach-ment of the dynamic stall vortex to the airfoilsurface (Fig. 14) and the renewed induction of alow pressure area {7}. These two characteristicscorrespond with the above mentioned minimumCl min = -0.002 and secondary maximum Cl max2

= 0.471 of the lift coefficient, occurring at14.82°↓ (t/T = 0.67) and 13.09°↓ (t/T = 0.70)respectively.

The phase of flow reattachment features asmoothened contour, as primary and secondaryvortex structures no longer have any effect onthe airfoil’s pressure distribution. With reattach-ment of the flow starting at the leading edge, arenewed build up of the primary suction peakcan be observed {8}. However, the completionof the downstroke leads to its reduction, as theincidence angle decreases and the initial state offlow is reached at the motion’s lower deadcenter.

3.2.3 ResumeComparing the numerical results with thecorresponding experimental test case acquiredby Ranke [9] showed good conformity in thephenomenology. The onset of the thin reverse

Alexander Pechloff

773.10

flow layer, the development of the dynamic stallvortex, post stall vortex shedding and flowreattachment occurred within ∆t/T ≈ ± 0.03 non-dimensional phase increments pertaining to theexperimental values. Simulated phenomenaduring the phase of aperiodic vortex shedding,such as secondary vortex structures, are in goodqualitative agreement with particle imagevelocimetry data obtained by Raffel [8] andWernert [10], using a somewhat modifiedincidence law. After FLOWer’s adaptation tothe incompressible flow regime, its ability toproperly predict the unsteady boundary layerseparation has been successfully evaluated andestablished.

In order to determine the effect of the timeincrement on the Navier-Stokes solution thepreviously used time step size is halved to ∆t/T= 0.005 and the computation of the pitchingcycle repeated. Consequently, the sinusoidalpitching motion is now discretized in 200individual time steps. Comparison of the liftcoefficients development over the incidence forboth cases, as depicted in Fig. 15, shows noinfluence of the time step size up to the pointwhere maximum lift occurs and post stall vortexshedding commences. Beyond this mark the liftcoefficient for the new computations does notreproduce, but nonetheless returns a curve withsimilar local characteristics, such as thesecondary lift maximum. This divergent be-havior can be attributed to the strong aperiodiceffects persistent throughout the vortex shed-ding phase, which are especially sensitive to thechosen time step.

4 Conclusions

By analyzing the behavior of the Navier-Stokessolver FLOWer in regards to its stabilityenhancing and convergence accelerating fea-tures, parameter settings for incompressibleflow conditions were determined. The obtainedsteady state solution was used as a starting pointfor unsteady simulations of the pitching NACA0012 airfoil. FLOWer’s ability to predict thephenomology of the two-dimensional deepdynamic stall process was validated by simu-lating an acquired experimental test case. The

results and insight gained through methodologyand calculations received further applicationduring quasi-three-dimensional computations ofthe unsteady boundary layer separation on aninfinite swept wing.

References

[1] Coton, F.N., Galbraith, R.A.McD., Jiang, D. andGilmour, R.: An experimental study of the effect ofpitch rate on the dynamic stall of a finite wing.Department of Aerospace Engineering, University ofGlasgow, Glasgow: G12 8QQ, 1996

[2] Jameson, A., Schmidt W. and Turkel, E.: Numericalsolutions of the Euler equations by finite-volumemethods using Runge-Kutta time-stepping schemes.AIAA 81-1259, 1981

[3] Kroll, N.: FLOWer - Installation and user handbookVersion 114/115. Deutsche Forschungsanstalt fürLuft- und Raumfahrt e.V., Institut fürEntwurfsaerodynamik, 1997-1998

[4] Kroll, N. and Jain, R.K.: Solution of two-dimensionalEuler equations - experience with a finite volumecode. Braunschweig: DFVLR-FB 87-41, 1987

[5] Martinelli, L. and Jameson, A.: Validation of amultigrid method for the Reynolds averagedequations. University of Princeton, Department ofMechanical and Aerospace Engineering: AIAA Paper88-0414, 1988

[6] Pechloff, A.: Numerische Simulation einesschwingenden und schiebenden NACA0012 Flügelsunter Anwendung des Navier-Stokes-Lösers FLOWer.Technische Universität München, Lehrstuhl fürFluidmechanik, Diplomarbeit, 1998

[7] Radespiel, R. and Rossow, C.: Efficient cell-vertexmultigrid scheme for the three-dimensional Navier-Stokes equations. Institute for Design Aerodynamics,Federal Republic of Germany: AIAA Journal, Vol.28, No. 8, 1990

[8] Raffel, M., Kompenhans, J., Wernert P.:Investigation of the unsteady flow velocity fieldabove an airfoil pitching under deep dynamic stallconditions. DLR, German Aerospace ResearchEstablishment, D-37073, Göttingen, Germany:Experiments in Fluids, Springer Verlag, 1995

[9] Ranke, H.: Instationäre Grenzschichtablösung inebener inkompressibler Strömung. TechnischeUniversität München, Lehrstuhl für Fluidmechanik,Dissertation, 1994

[10] Wernert, P., Geissler, W., Raffel, M. andKompenhans J.: Experimental and numerical inves-tigations of dynamic stall on a pitching airfoil. DLR,German Aerospace Research Establishment, D-37073, Göttingen , Germany: AIAA Journal, Vol. 34,No. 8, 1996

773.11

NUMERICAL SIMULATION OF A PITCHING NACA 0012 AIRFOIL

Figures

-10.00 -5.00 0.00 5.00 10.00x

-10.0

-8.0

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

8.0

10.0

y

Fig. 1. 384 × 96 ‘C-topology’ grid

-0.25 0.00 0.25 0.50 0.75x

-0.2

-0.1

0.0

0.1

0.2

y

Fig. 2. Mesh structure in airfoil proximity

Relationship between CFL-number and smoothing paramater εxyz:

α = 0°, Ma = 0.1, Re = 2.5 x 10-5, 4 level simple multigrid, w- sym typedissipation evaluation on first RK-stage: (5,1)-schemeimplicit residual averaging applied on all RK-stages

0

100

200

300

400

500

600

700

800

900

1000

1100

1200

1300

1400

1500

0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2εxyz

mul

tigr

id c

ycle

s

CFL 3.75

CFL 5.625

CFL 7.5

Residual tolerance:

||dρ/dt||2 ≤ 10-5 within 1500 multigrid cycles

Grid 384 x 96,2D calculations on HP-Workstation

Boundary of convergence:CFL = 3.75 : 0.55 < εxyz < 1.0

CFL = 5.625: 0.7 < εxyz < 1.7

CFL = 7.5 : 1.3 < εxyz < 2.2

Optimal convergence:CFL = 3.75 : min. cycles 404 @ εxyz = 0.6 [0%]

CFL = 5.625: min. cycles 379 @ εxyz = 1.2 [-6%]

CFL = 7.5 : min. cycles 368 @ εxyz = 1.9 [-9%]

Fig. 3. CFL-number dependent boundaries ofconvergence

0 50 100 150 200 250 300 350 400 450 500 550 600

multigrid cycles

3.0E-05

6.0E-05

9.0E-05

1.2E-04

1.5E-041.8E-042.1E-042.4E-04

den

sity

resi

dual

xyz constant [1.9]

xyz adaptive [1.9 - 1.9]

xyz adaptive [1.8 - 2.0]

xyz adaptive [1.7 - 2.0]

Alpha = 0 [deg], Ma = 0.1, Re = 2.5 x 105, 4 level simple multigrid (w-sym), cfl 7.5,dissipation evaluation only on first RK stage, k(4) = 1/32

Case designation: test3_cfl75_eps_*

Comparison between constant and adaptive residual averaging:

Residual tolerance: 1 x 10-5

Calculated on 384 x 96 gridwith FLOWer 114.8 [HP]

constant [1.9]adaptive [1.9 - 1.9] adaptive [1.7 - 2.0]

adaptive [1.8 - 2.0]

Required accuracy obtained in ...

xyz constant [1.9] : 368 cycles

xyz adaptive [1.9 - 1.9] : 365 cycles

xyz adaptive [1.8 - 2.0] : 391 cycles

xyz adaptive [1.7 - 2.0] : 568 cycles

εεεε

ε

εεε

Fig. 4. Comparison of the residual averaging techniques

500 1000 1500multigrid cycles

10-5

10-4

10-3

10-2

10-1

100

den

sity

resi

du

al

k(4) = 1/16k(4) = 1/32k(4) = 1/64k(4) = 1/128k(4) = 1/256k(4) = 1/512

Case designation: test3_cfl75_eps19_a2_r4_*

Influence of third order dissipation constant k(4) on convergence history (2D calc.):Alpha = 0 [deg], Ma = 0.1, Re = 2.5 x 105, CFL = 7.5, QFILS 1 0 0 0 0, SMS = 1 1 1 1 1, BETA 1 1 1 1 1I2D3D 2, IVIS 40, xyz = 1.9 (constant), k(2) = 1/2, 4 level simple multigrid (w-sym type)

Abrupt boundary of convergence exits for k(4) > 1/16

Calculated on 384 x 96 gridwith FLOWer 114.8 [SP2]

ε

Required accuracy obtained in ...k(4) = 1/16 : 298 multigrid cyclesk(4) = 1/32 : 365 multigrid cyclesk(4) = 1/64 : 453 multigrid cyclesk(4) = 1/128 : 567 multigrid cyclesk(4) = 1/256 : 818 multigrid cyclesk(4) = 1/512 : 1285 multigrid cycles

Residual tolerance: 1 x 10-5

Fig. 5. Influence of third order background dissipation

Alexander Pechloff

773.12

1.401.451.50

1.701.90

2.102.15

1/16

1/64

1/2560

250

500

750

1000

1250

1500multigrid

cycles

εxyz

k(4)

Relationship between smoothing parameter εxyz und third order dissipation

constant k(4): Area of attainable convergence

1250-1500

1000-1250

750-1000

500-750

250-500

0-250

Residual tolerance:

||dρ/dt||2 ≤ 10-5 within 1500 multigrid cycles

Grid 384 x 96, 2Dcalculations on SP2

Fig. 6. Convergence area obtainedfor εxyz and k(4) variation

0.060.03

0.020.01

0.00

2.11.

91.71.

51.45

0.0122

0.0124

0.0126

0.0128

0.0130

0.0132

0.0134

Cd

k(4)εxyz

Relationship between drag coefficient Cd and third order dissipation

constant k(4) (εxyz Variation)

Residual tolerance:

||dρ/dt||2 ≤ 10-5 within 1500 multigrid cycles

Grid 384 x 96,2D calculations on SP2

Cd-Deviation (@ εxyz = 1.9):

k(4) = 1/16 : Cd = 1.330·10-2 [+6.49%]

k(4) = 1/32 : Cd = 1.249·10-2 [+0.00%]

k(4) = 1/64 : Cd = 1.251·10-2 [-0.16%]

k(4) = 1/128: Cd = 1.289·10-2 [-3.20%]

k(4) = 1/256: Cd = 1.332·10-2 [6.64%]

k(4) = 1/512: Cd = 1.368·10-2 [9.53%]

Fig. 7. Influence of k(4) on the drag coefficient Cd

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0alpha [deg]

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

Cl

cycle 1cycle 2cycle 3cycle 4cycle 5cycle 6

Pitching NACA 0012, Ma = 0.1, Re = 2.5 x 105, alpha 0 - 20 [deg]:

Lift coefficent history for cycles 1 through 6

Significant Cl data for cycle 6:

Cl max = 2.015 (19.69 [deg] , t/T = 0.54)

Cl min = -0.002 (14.82 [deg] , t/T = 0.67)

Cl max2 = 0.471 (13.09 [deg] , t/T = 0.70)

Cl max2

Cl max

Cl min

upstrokedown-

stroke

= 0 [deg]kred = 0.30 β

∆

Evolution of the dynamic stall process:

A : (-> 19.30 [deg] , t/T = 0.44) Attached flow

B : (-> 19.30 [deg] , t/T = 0.56) Developement of dynamic stall vortex

C : (-> 11.87 [deg] , t/T = 0.72) Poststall vortex shedding

D : (-> 0.00 [deg] , t/T = 1.00) Flow reattachment

C

A

D

B

2D calculation, t/T = 0.01

onset of thin reverse-flow

layer (10.63 [deg] , t/T = 0.26)

↓↓

↓↓↓

↓

↓

↓

Fig. 8. Lift coefficient over incidence(kred = 0.30, ∆t/T = 0.01)

-2.0-1.00.01.02.03.04.05.06.07.08.09.0

10.0

-Cp

-0.30-0.15

0.000.15

0.300.45

0.600.75

x/c0.0

2.55.0

7.510.0

12.515.0

17.520.0

alpha [deg]

Y

Z

X

-Cp

9.75

9.00

8.25

7.50

6.75

6.00

5.25

4.50

3.75

3.00

2.25

1.50

0.75

0.00

-0.75

-1.50

Pitching NACA 0012 wing: Variation of

chordwise pressure with angle of attack

Ma = 0.1

Re = 2.5 x 105

alpha 0 - 20 [deg]

kred = 0.30

= 0 [deg]β

t/T = 0.01

2D calculation

∆

t/T = 0.01 - 0.50

2

1(6th cycle, upstroke, upper surface)

Fig. 9. Cp-α-x/c-contour (kred = 0.30, upstroke)

-2.0-1.00.01.02.03.04.05.06.07.08.09.0

10.0

-Cp

-0.30-0.15

0.000.15

0.300.45

0.600.75

x/c0.0

2.55.0

7.510.0

12.515.0

17.520.0

alpha [deg]

Y

Z

X

-Cp

9.75

9.00

8.25

7.50

6.75

6.00

5.25

4.50

3.75

3.00

2.25

1.50

0.75

0.00

-0.75

-1.50

Pitching NACA 0012 wing: Variation of

chordwise pressure with angle of attack

Ma = 0.1

Re = 2.5 x 105

alpha 0 - 20 [deg]

kred = 0.30

= 0 [deg]β

t/T = 0.01

2D calculation

∆

t/T = 0.51 - 1.00

3

6

54

7

8

(6th cycle, downstroke, upper surface)

Fig. 10. Cp-α-x/c-contour (kred = 0.30, downstroke)

773.13

NUMERICAL SIMULATION OF A PITCHING NACA 0012 AIRFOIL

-0.50 -0.25 0.00 0.25 0.50 0.75 1.00x

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

y

utot: -0.258 -0.214 -0.169 -0.125 -0.081 -0.036 0.000 0.041 0.086 0.130 0.174 0.219

t/T : 0.440alpha: 19.298

Fig. 11. Flow field (kred = 0.30, α = 19.30°↑ , t/T = 0.44)

-0.50 -0.25 0.00 0.25 0.50 0.75 1.00x

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

y

utot: -0.258 -0.214 -0.169 -0.125 -0.081 -0.036 0.000 0.041 0.086 0.130 0.174 0.219

t/T : 0.470alpha: 19.823

Fig. 12. Flow field (kred = 0.30, α = 19.82°↑ , t/T = 0.47)

-0.50 -0.25 0.00 0.25 0.50 0.75 1.00x

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

y

utot: -0.258 -0.214 -0.169 -0.125 -0.081 -0.036 0.000 0.041 0.086 0.130 0.174 0.219

t/T : 0.560alpha: 19.298

Fig. 13. Flow field (kred =0.30, α = 19.30°↓ , t/T = 0.56)

-0.50 -0.25 0.00 0.25 0.50 0.75 1.00x

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

y

utot: -0.258 -0.214 -0.169 -0.125 -0.081 -0.036 0.000 0.041 0.086 0.130 0.174 0.219

t/T : 0.670alpha: 14.818

Fig. 14. Flow field (kred = 0.30, α = 14.82°↓ , t/T = 0.67)

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0alpha [deg]

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

Cl

cycle 6cycle 7

Pitching NACA 0012, Ma = 0.1, Re = 2.5 x 105, alpha 0 - 20 [deg]:

Lift coefficent history for cycle 7 (in comparison with cycle 6)

Significant Cl data for cycle 7:

Cl max = 2.061 (19.51 [deg] , t/T = 0.550)

Cl min = 0.031 (14.82 [deg] , t/T = 0.670)

Cl max2 = 0.640 (13.09 [deg] , t/T = 0.700)

Cl max2

Cl max

Cl min

upstroke down-

stroke

= 0 [deg]kred = 0.30 β

∆

C

A

D

B

2D calculation, t/T = 0.005

Evolution of the dynamic stall process:

A : (-> 19.30 [deg] , t/T = 0.44) Attached flow

B : (-> 19.30 [deg] , t/T = 0.56) Developement of dynamic stall vortex

C : (-> 11.87 [deg] , t/T = 0.72) Poststall vortex shedding

D : (-> 0.00 [deg] , t/T = 1.00) Flow reattachment

onset of thin reverse-flow

layer (11.25 [deg] , t/T = 0.27)

↓↓↓

↓

↓

↓↓↓

Fig. 15. Lift coefficient over incidence(kred = 0.30, ∆t/T = 0.005)