euler flutter analysis of airfoils using unstructured dynamic meshes

8/12/2019 Euler Flutter Analysis of Airfoils Using Unstructured Dynamic Meshes

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436 J. A I RCRA FT V O L . 27, NO. 5

Euler Flutter Analysis of AirfoilsUsingUnstructured Dynamic MeshesRuss D . Rausch*Purdue University, West Lafayette, Indiana 47907JohnT.BatinatNASA Langley Research Center, Hampton, Virginia 23665-5225an dHenry T. Y. YangJPurdue University, West Lafayette, Indiana 47907

Modificationsto a two-dimensiona l, unsteady Euler code for theaeroelasticanalysis of a irfoils are described.Th emodificationsinvolveincludingth estructural equationsof motiona nd their simultaneous t ime integrationwith th e governingflow equations. A novel aspect of the capabilityi s that th e solutions ar e obtained usingunstructuredgrids madeup oftriangles. Comparisonsar emadewithparallel calculations perform ed using lineartheoryand a structured gridEulercode to assess the accuracy of the un struc ture d grid Euler results. Results arepresented for a flat-plate airfoil and the NACA0012 airfoil to demo nstrate applications of the Euler code forgeneralized force computations an d aeroelastic analysis. In these comparisons, tw o different finite-volumediscretizations of the Euler equations on u nstruc tured meshes were employed. Sensitivity of the Euler results tochanges innu merical parameters was also investigated.

NomenclatureAy =generalized aerodyn amic force resulting from pressureinduced by mode j acting through mode /a - nondimensional distance from midchordtoelastic axis 0 0 = freestream speed of soundb = airfoil semichord,c/2Cp = pressure coefficientc = airfoil chord j = generalized damping of mode /k - reduced frequency,^c/2Uoakt =generalized frequen cyof mode /M O O = freestream Mach numberm - airfoil massperunit spanmi = generalized mass of mode /Q =nondim ensional dynam ic pressure [(7oo/(^waV/I)]2Q f = nondimensional dynamic pressure at flut terQi = generalized force in mode /Q i =generalized displacement of mode ira =airfoil rad ius of gyration about elastic axis5 = Laplace transf orm variable, o+ / o >T =t ime, st =time, nondimensionalized by freestream speed ofsound andairfoil chord,Tajc

Received Feb. 19, 1989; presented as Paper 89-1384 at the 30thStructures, Structural Dynamics,an d Materials Conference, Mobile,AL, April 3-5, 1989. Co pyrig ht 1989by theAmerican InstituteofAeronautics an d Astronautics, Inc. No copyright is asserted in theUnitedStates under Tit le 17, U.S. Code. The U .S. Go vernment has aroyalty-free licenseto exerciseal l rights under th e copyright claimedherein fo rGo vernm ental purposes. A llother rightsar ereservedby thecopyright ow ner.*Gradua te Research Assistant, School of Aeronauticsan d Astro-naut ics . M ember AIAA.tSenior R esearch Scientist, U nsteady Aerodynamics B ranch, Struc-turalD ynamics Division. Senior M ember AIAA .^Professor an d Dean, Schools of Engineering. Fellow AIAA ,

t / o o = streamwise freestream speedii j = load vectorX j = state vectorxa - nondimensional distance from elastic axistomass centero i Q =mean angle of attackACP = unsteady lifting pressure coefficient normalized by theampli tude of oscillation9 =integralof state-transition matrixj =airfoil mass ratio,m/^p^b2p o o = freestream densityo = real part of Laplace transform variable = state-transition matrixw = angular frequencyW f = uncoupled natural frequency of bending modec o a = uncoupled natural frequency of torsion mode

IntroductionCONSIDERABLE progress has been made over th e pastdecade on developing computer codes for transonicaeroelastic analysis.1This research ha sbeen high ly focusedo ndeveloping finite-difference codes for the solution of the tran-sonic small-disturbance2'3 and full-potential4 '5 equations, al -though efforts are currently underway at the higher equationlevelsaswell.6'10Th e above work on aeroelastic analysis withthe Euler and Navier-Stokes equations has all been done withflow solvers that assume that the computational grid has anunderlying geometrical structure. The applications havealsobeen limited to simple geometries such as airfoils an d wings.Several problems arise, however, in extending these proce-durestotreat more complicated geometries. For example,it isgenerally recognized thatit isdifficult togenerate astructuredgrid about a complete aircraft configuration. Other diffi-culties arise in moving the grid for aeroelastic motion whereth e grid must conform to the instantaneous shape of thevehicle. Therefore, an alternative approach that ca n treatcomplex aircraft configurations undergoing realistic deforma-tions an d motions is desired.A n alternative approach for the modeling of complex air-craft configurations is the use of unstructured meshes.11 14 Intwo dimensions these meshes are constructed from triangles


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M AY 1990 U N S T R U C T U R E D D Y N A M I CMESHES 43 7and in three dimensions they consist of an assemblage oftetrahedra.Thesetriangles or tetrahedra are oriented to con-form to the geometry being considered, thus making possibleth e treatment of complicated shapes. Applications of thesemethodshave beendemonstratedfor multielementairfoils11 '12an dcompleteaircraft configurations.13 '14 Afurther advantageof the unstructured grid methods is that they easily permitmesh ref inement in regions of high-flow gradients to moreaccurately resolve the physics of the f low.12 '15 '16A nassessment of theapplicability of theunstructured gridmethodology for unsteady aerodynamic analysis of airfoilswasreported inRef . 17whereanunsteady Euler solverbasedon an unstructured grid of triangles was developed to treatoscillating airfoils. The capability included a dynamic meshalgorithm to allow for airfoil motion. The algorithm isquitegeneralinthat it can treat realisticmotions and deformationsof multiple two-dimensionalbodies.The purpose of this study was to extend the two-dimen-sional,unsteady Eulercode of Ref . 17to the aeroelastic anal-ysis of airfoils. This represents a first step toward aeroelasticanalysis of aircraft using unstructured meshes.18 Results ar epresented for a flat-plate airfoil and for the NACA 0012airfoilto dem onstrate applications of the Euler code togener-alizedaerodynamic force computat ion an d aeroelasticanaly-sis. Detailed comparisons ar emade for resultsobtainedusingthe two-dimensional Eulercode, the CFL3D19 Euler /Navier-Stokescode,a ndlineartheory.20Thesecomparisons permit anassessmentof theaccuracyof theEuler codefor suchapplica-tions. In thesecomparisons, tw o different finite-volume dis-cretizationsof theEulerequationson unstructuredmeshesa reemployed.Sensitivityof theEulerresults tochangesin numer-ical parameters was also investigated.

EulerSolution AlgorithmThe two-dimensional, unsteady Euler equations are solvedusing the multistage Runge-Kutta, time-stepping scheme of

Ref. 17.This algorithmusesafinite-volumespatialdiscretiza-tion forsolutionon an un structu red grid made up of triangles.The original algorithm of Ref. 17 was anode-based schemewhereby the flow variables are stored at the vertices of thetriangles. A second algorithm, a cell-centered scheme, wasalso employed in the present study. This second scheme isbased on unpublished work of the second author. In thecell-centered scheme, th e flow variables are stored at the cen-troids of thetriangles. Inbothalgorithms, artificial dissipa-tion is added explicitly to prevent oscillations near shockwaves and to damp high-frequency uncoupled error modes.Specifically, an adaptive blend of harmonic an d biharmonicopera tors is used, corresponding to second- and fourth-differ-encedissipation, respectively.The biharmonicoperator pro-videsa background dissipation todamphighfrequencyerrors,and the harmonic operator prevents oscillations near shockwaves. The algorithms also employ enthalpy damping, localtime stepping, and implicit residual smoothing to accelerateconvergence to steady state. The local time stepping uses th emaximum allowable step sizeat eachgrid point for thenode-based scheme and for each triangle in the cell-centeredscheme, as determined by a local stability analysis. The im-plicit residual smoothing permits the use of local time stepsthat are larger than those imposed by the Courant-Fredricks-Lewy stability condition. This is achieved by averaging theresiduals implicitly with neighboring values. A time-accurateversion of the residual smoothing isalso used for global t imestepping during unsteady applications of the code.Withrespect to boundary conditions,acharacteristic analy-sis based on Riemann invariants is used to determine th evaluesof the flowv ariables on theouter boundaryof thegrid .This analysis correctly accounts for wave propagation in thefa rfield, which is important fo r rapid convergence to steadystate and serves as a nonreflecting boundary condition forunsteadyapp lications.

Dynamic Mesh AlgorithmA dynamic mesh algorithm is used to move the mesh fo runsteady airfoil calculations. The method , as developed inRef. 17, models the triangulated mesh as a spring networkwhere each edge of each triangle represents a spring withstiffness inversely proportional to the length of that edge. Inthis procedure, grid points along th eouter boundary of themesh are held fixed, and the instantaneous locations of thepointson the airfoil (inner boundary)ar e specified. For aero-elastic calculations, th e position of the inner boundary isdetermined by the structural equations of mot ion . The loca-tionsof theinterior nodesare then determined bysolvingth estaticequilibrium equations that result from a summation offorcesat each nod e inboththexa ndydirections.T hesolutionof the equilibriumequations isapproximated by using a pre-dictor-corrector procedure, which first predicts the newloca-tions of thenodesbyextrapolationfrom gridsat previoustimelevelsand thencorrects these locationsbyusing severalJacobiiterations of the static equilibrium equations. The predictor-correctorprocedureis relatively efficient because only one ortw oJacob i iterationsarerequiredtomoveth emeshaccurately.

Pulse TransferFunction AnalysisGeneralizedaerodynamicforces,whicha retypically usedinaeroelastic analyses, may be obtained by calculating severalcycleso f aharm onicallyforced oscillationwiththedetermina-tion of the forces based on the last cycle of oscillation. Th emethod of harmonic oscillation requires one calculation fo reach value of reduced frequency of interest. By contrast,generalized forces may be determined for a wide range ofreduced frequency in asinglecalculationby thepulsetransferfunction analysis.21 '22 This, of course, relies on the local lin-earity of the airloads with respect to mode and amplitude ofmot ion ,whichmust beassuredfor reliable use of the metho d.Time-Marching Aeroelastic Analysis

In this section th e aeroelastic equations of motion, th et ime-marching solution procedure, and the modal identifica-tion technique are described.Aeroelastic Equations of Motion

The aeroelastic equationso f motioncan bewri t ten fo reachmode /asC M + k&i = Qi (1)

where qt is the generalized normal mode displacement, m; isthe generalized mass, c/ is the generalized damping, A :/ is thegeneralizedstiffness, an d Q /is thegeneralizedforcecomputedby integrating the pressure weighted by the mode shapes.These equations of motion are derived by assuming that th edeformation of thebodyunder considerationcan be describedby a separation of variables involving the summation of freevibration modes weighted by generalized displacements. Thisanalysis differs from th e traditional form of the two-dimen-sionalaeroelasticequations of motion,whichar eusuallywrit-ten in terms of plunge and pitch degrees of freedom. Thisnormal mode form of the equations was implemented in thepresent studydue to therelativeeaseo fextendingthemethod-ologyt o three-dimensional structureswherethe normal moderepresentation is most common.Time-Marching SolutionThe solution procedure for integrating Eq. (1) issimilar tothat described byEdwards etal.23'24 A similar formulation isimplementedin thepresentstudy formultiple degreesof free-dom. Here the linearstateequations arewrittenas

X i = A X J + Bui (2)where A an d B ar e coefficient matrices that result from th echangeof variables# / = [ < / / < ? / ]r , and u is the nondimensional


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43 8 RAUSCH, B A TIN A , A N D Y A N G J. A I R C R A F Tgeneralized force weighted by mode /. Equation (2) is inte-gratedi ntimeusing the modifiedstate-transitionmatrixstruc-turalintegra tor24 implementedas a predictor-corrector proce-dureaccording toPredictor:

Y * l */ ~ ~Corrector:

(3a)

(3b)where is thestate-transitionmatrix,and 9 theintegralof thestate-transition matr ix from time step n to n + 1.Modal Identification Technique

Damping and frequency characteristics of the aeroelasticresponsesare estimated from th e responsecurves by using themodal identification technique of Bennett an d Desmarais.25The modal estimates ar e determinedby a least-squares curvefit of the responses of the formqt(T) = a0+ EeaJT[a j cos(w/r) + i = 1,2 (4)

where m is the number of modes which was selected to beequal to two in the present study.State-Space Aeroelastic StabilityAnalysis

Asan alternative to thetime-marching analysis, aeroelasticstabilityanalyseswereperformed usingstate-space aeroe lasticmodeling as developed in Refs . 26 and 27.This locally linearaeroelastic stability analysis is relatively inexpensive and re-tainsthenonlinearpropertiesof thetransonic mean flow. Themodeli sderivedbycurv e fit t ing th egeneralized force transferfunctions obtained from transient responses resulting frompulsedmotions of theairfoil modes. TheseFade approximat-ing functions26 may be written as a set of ordinary differen-tial equations and coupled to the airfoilequations ofmotion.This leadsto alinearfirst-ordermatrixequation,whichcan besolved using linear eigenvaluesolution techniques.The result-in g eigenvalues are plotted in a dynamic pressureroot-locustype format in the complexs plane.

.8 1.0

a plunge motion

Linear theoryo Euler

Fig. 1 Partial view of 64 x 32 grid of triangles about the flat-plateairfoil.

b pitch motionFig. 2 Comparisons of unsteady pressure distributions computedusing linear theory and the node-based Euler algorithm for the flat-plate airfoil undergoing harmonic motion at Moo =0.5, ao= 0 deg,an dk =0.25.

Results andDiscussionToassesstheaccuracyof the two-dimensional,unstructuredgrid Euler code for generalized force computation an d aero-elastic analysis, calculations were performed for a flat-plateairfoiland for theNAC A 0012airfoil. Comparisonsare madewith parallel calculations performed using alternative meth-ods.Generalizedaerodynamicforces werecomputedusingth epulsetransfer-function analysis aswellas bysimpleharmonicoscillation fo r several values of reduced frequency. In theharmonic calculations, th e airfoil was forced to oscillate fo r

three cyclestoobtaina periodic solutionusing 2500steps/cy-cleo fmotion. Bothplunge and pitch-about-the-quarter-chordmotions were considered. The amplitudes of oscillation were0.01 chord lengths and 0.1 deg for plunge and pitch, respec-tively. Aeroelastic results are presented for a much studiedconfiguration designated as case A in Ref. 28 , which ha snormalmodessimilarto those of astreamwisesectionnearth etip of a swept-back wing. The wind-off bending and torsionfrequenciesare 71.33 and535.65rad/s,respectively.The pivotpoint for the bending mode is located 1.44 chord lengthsupstream of theleadingedge of theairfoil. Thepivotpointf orthetorsion modeis0.068chord lengths forwardof midchord.These mode shapes an d natural frequencies weredeterminedby performing a free vibration analysis with th e aeroelasticequations of motionwritten in the traditional form using th efollowing structural parameter values: a= -2.0, xa= 1.8,ra= 1.865, a? *= 10 0 rad/s, and wa = 100 rad/s. Also, theairfoil massratiowas /*= 60. Generalized displacements cor-responding to the inertially coupled modes are defined as q\and < 7 2 respectively. Initialconditions for the time-marchingaeroelastic analysis were #i(0)=2 and 2(0 )=0-01-


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MAY 1990 U N S T R U C T U R E DD Y N A M I CMESHES 439Flat-Plate Airfoil Results

Calculationswereperformed for a flat-plateairfoil toassessthe accuracy of theu ns tructured grid Eulercode for a simpleconfiguration whereahighly accurate linear theorysolutioni savailable fo r comparison. The results wereobtained using amesh oftriangles,whichwasgeneratedanalytically by amod-ified Joukowskit ransformat ion. Apartialviewof the mesh isshown inFig. 1. Th e gridhas 2048 nodes, 4096triangles, an dextends 20chordlengths from the airfoil witha circularouterboundary. Also,thereare 64pointswhichlie on thesurfaceofthe airfoil . Calculations were performed for the flat-plateairfoil at a freestream Mach number of M & =0.5 and 0 degangle of attack. Results were obtained using both th e node-based and thece ll-centered spatialdiscretizationsof theEulerequations. Comparisons ar e made with computat ions per-formed using th e linear theory program of Ref. 20, which isbased on thek ernel-function method.Unsteady Pressure Comparisons

Unsteady pressure distributions due to simple harmonicmotion were computed for the flat-plate airfoil to determinethe accuracy of the pressures by making comparisons withlinear theory results. The Euler results shown were obtainedusing the node-based discretization of the Euler equat ions .The calculationswereperformed at areducedfrequency basedon semichord of k =0.25 for both plunge and pitch-about-the-quarter-chord motions. TheseresultsarepresentedinFig.2 as real and imaginary components of the lifting pressurecoefficient normalized by the amplitude of oscillation ACP .The comparisons between Euler and linear theory pressuredistributions indicategood agreementbetween the twosetsofresults, which tend to verify th eEuler calculations. In thesecomparisons th e real parts of th e pressure distributions agreeslightlybetterthan the imaginaryparts. Also, the largest dif-ferences occur in the imaginary part of the unsteady pressuredue to plunge (Fig.2a) which isunderpredicted in comparisonwith linear theory . Furthermore, a fundamental differencebetween the Euler and linear theory pressure distributions isth efactthatthe pressure jum p at the leading edge iszeroin theEuler calculation with th e node-based scheme in comparisonwith infinity predicted by linear theory. The Euler pressurejump iszeroat th eleading edge since the grid pointhere canbe defined as either an upper surface or lower surface pointand hence AC ^ = 0. According to linear theory, however,therei s asquare-root singularity in the lifting pressurecoeffi-cient at the leading edge. Consequently, some of the differ-ences in the Eulerpressures on thesurface of theairfoil aft ofthe leading edge and subsequent differences in generalizedforces in comparison with linear theory may beattributed tothis fundamental difference in the character of the solutionsnear the leading edge.Generalized Force ComparisonsGeneralizedaerodynamicforcesfo r the flat-plateairfoilarepresented in Fig. 3; they ar e computed using three differentmethods including the pulse analysis, the harmonic analysis,and linear theory. Here, both sets of Euler results were firstobtained using the node-based spatial discretization of theEulerequations. The resultsareplottedasreala nd imaginarycomponents of the unsteady forces A\j as a function of re-duced frequency k . In this section, plunge and pitch-about-the-quarter-chord ar e defined as modes 1 and 2, respectively.Thus, for example,Auis the lift coefficient due to pitching.The harmonic results wereobtained at five values of reducedfrequency:k =0.0,0.25,0.50, 0.75,and1.0.Theseresultsareused to determine the accuracy of the results obtained usingthe pulse analysis. Both of these sets of results arecomparedwith linear theory results, which are thecorrect solution fo rthiscase.As should be expected, theEuler forces computedusingthepulse analysis are in excellent agreement with the forces corn-

L inear theory Euler pulseO Euler harmonic

.8 1.0

Fig. 3 Comparisons of generalized aerodynamic forces computedusing linear theory and the node-based Euler algorithm for the flat-plate airfoil at Moo = 0.5 and o=0 deg.

puted using simple harmonic motion at all values of reducedfrequency that were considered. Both of these sets of forcesagree reasonablywellwith thelineartheoryforcesat thelowervalues ofreducedfrequency, althoughdifferences occurat thehigher values of reduced frequency. For example, for the liftcoefficient due toplungeAn theimaginarypart isunderpre-dictedby approximately 20% atk - 1.0 incom parison withlinear theory. Also, the forces due to pitch compare slightlybetter than th e forces due to plunge.To investigateth esourceof the discrepanciesbetween Eulerand linear theory forces, many numerical parameters in thenode-based algorithm were varied. These variations include:1)meshrefinement, 2) the amplitudeo fmotion,3) howmanyJacobiiterationswereused in thedynamic meshcapability, 4)whether the mesh was moved rigidly following th e airfoilmotion or with the dynamic mesh capability,5) the temporalorder of accuracy in the numerical determination of gridspeeds, 6) whether the grid speeds were determined numeri-cally or analytically, 7) the level of explicit artificial dissipa-tion that was used in the calculation, and 8) whether or notimplicitresidualsmoothingw asusedto increase the allowablestep size. In all cases, th e results compared to plotting accu-racy and are thus considered to beconverged with respect tothesev ariations.The unsteady Euler calculations for the flat-plate airfoilwerethen repeated using the cell-centered algorithm to deter-mine theeffects of thespatialdiscretization onth egeneralizedforces. Theseresults and comparisonsar epresented inFig.4.Allof theconditionsthatwereusedtodeterminetheseresultswere thesameasthoseused toobtain the results ofFig. 3. Incontrast with th e node-based Euler results, the cell-centeredEuler forcesare inverygoodagreementwithlineartheory,f orbothplungeand pitchmotions,for the entire range ofreducedfrequency that was considered. Also, th e real parts of the


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440 RAUSCH, BATINA, AND Y A N G J. AIRCRAFTLinear t h e o r y E u l e r - p u l s e

P E u l e r - h a r m o n i c

8 1.0

Fig. 4 Comparisons of generalized aerodynamic forces computedusing linear theory and the cell-centered Euler algorithm for theflat-plate airfoil atM = 0.5 and X Q = 0deg.

forces compareslightlybetterthanthe imaginaryparts.There-fore, because of the improved accuracy provided by the cell-centered discretization, the remainder of the study was con-ducted using the cell-centered scheme.Aeroelastic Comparisons

Aeroelastic results for case A were obtained for the flat-plate airfoil using both th e state-space and time-marchinganalyses. The state-space calculations were performed usingth e generalized aerodynamic forces determined by the Eulerpulseanalysis and by lineartheory. Theseresults are presentedinadynam ic pressureroot- locus-type format in thecomplexsplane. The t ime-marching calculations were performed fo rthree values of dynamic pressure, which bracket th e fluttercondition,as de termined by the state-space analysis. Dampingan d frequency estimates were determined from the resultingaeroelastic transientsusingthemodalidentification technique.These estimates were then compared with values obtainedfrom th e state-spacecalculations to assess the accuracy of theEuler time-marching aeroelastic procedures. A quadratic in -terpolation of the t ime-marching dominan t damping valueswasalsoused to determine the flutter condition,whichcorre-sponds to z ero damping.A comparison of nondimensional dynamic pressure rootloci for the flat-plate airfoil ispresented inFig. 5. Theserootloci indicate that the bending- and torsion-mode frequenciesapproacheach otherfor low values of Q.Theflutterconditionis givenby theo=0 crossing of the bendingroot locus,whichcontinues into the unstable right-half plane as the dynamicpressure is increased fur ther . The two sets of root loci agreewell with ea_ch other , and the values of dynamic pressure atflutter are Q f =4. 8 and 4.6, as predicted by the Euler andlinear theory analyses, respectively.

_ Time-marching aeroelasticcalculationswere performed fo rQ=3.0,4.0,and5.0.Fromthesec alculations,thegeneralizeddisplacement of the second coupled mode < ?2 as a function oft imeTispresentedi nFig.6. The two-modecurvefits from th emodalidentification techniquea realsogiven. Note theagree-ment of the curve fits to thedata to withinplotting accuracy.These displacement responses showsub critical, nearneutrallystable, and unstable aeroelastic behavior corresponding toQ- 3.0, 4.0,and5.0,respectively, whichisconsistentwiththestate-spaceresults of Fig. 5.The component modes from th e two-mode curve fit of thedisplacementresponsesar e shown inFig.7 for all threecasesthat were considered. These component modes consist of adominant modeo f larger amplitudecorrespondingtobending(mode 1) and a second, smaller amplitude, damped mode

01

O Linear theoryn Euler

Bending

.2 -.8 -A 0 .4 .8

Fig. 5 Comparisons of nondimensional dynamic pressure root locifor the flat-plate airfoil atM = 0.5 ando= 0 deg for case A.Data

.02 .08 .1004 .06TFig. 6 Effects of nondimensional dynamic pressure on the gen-eralized displacement of the second coupled mode for the flat-plateairfoil atM = 0.5 and o= 0 deg for case A.

Amplitude

Amplitude

Amplitude

.01|0

.01.01

0.01.03.02.01

0.01.02

Mode 2 ~ /~XJS/ -Qs V / \ / Q

=

i i i i i .02 .04 .06 .08 .10T

Fig. 7 Effects of nondimensional dynamic pressure on thecomponentmodesof the aeroelastic system for the flat-plateairfoilatM o o = 0.5 ando=0 deg for case A .


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MAY 1990 UNSTRUCTURED D Y N A M I CMESHES 441Table 1 Comparisons betweenaeroelastic solutions for theflat-plate airfoil at x 0.5 and ao = 0 deg forcase A

Mode 1 Mode 2Method

3.0

4.0

5.0

Linear theoryEuler state-spaceEuler t ime-marchingLinear theoryEuler state-spaceEuler time-marchingLinear theoryEuler state-spaceEuler time-marching

-0.095-0.124-0.075-0.061-0.099-0.024

0.082 ;0.049 :0 . 1 2 1 :

.578.574.587

.913.921

.9182.3081.339L291

-0.462-0.471-0.465-0.639-0.663-0.591-0.937-0.999-0.815

4.6584.6654.6534.3414.3484.3233.9513.9554.037

Fig.8 Partial view of unstructured grid of triangles about theN A C A0012airfoil.

corresponding to torsion (mode 2). Damping and frequencyestimates of the two modes from th e curve fits are comparedwiththestate-space modelvaluesfrom the Euler pulse analysisan d from linear theory inTable 1.These comparisonsindicatethatthe time-marching valuescorrelatewell withtheothertwosets of results, which tends to verify th eEuler time-marchingaeroelastic procedures. In general, however , th e frequencyvaluesagree betterthan the dampingvalues,as expected.Thevalue of dynamic pressure at flutter from the time-marchinganalysiswas determined to beQf =4.4,whichiswithin5%ofth e linear theory flutter value .NACA 0012 Airfoil Results

Calculations were performed for theNACA 0012airfoil toassess th e accuracy of the unstructured grid Euler code fo rtransonic applications. The results were obtained using anunstructuredgrido ftriangles,apart ialviewo fwhichi sshowninFig.8. Thegridhas 3300nodes,6466triangles,andextends20 chord lengths from the airfoil with a circular outerboundary.Alsotherear e 110pointsthat lie on the surface ofth eairfoil/Calculations were performed for the airfoil atMO, = 0.8 and 0 deg angle of at tack. Comparisons are madewithparallel computations performed using theCFL3Dcode,ru nin a two-dimensionalmode.T heCFL3Dcode is anEuler/Navier-Stokescode based on cell-centered, upwind-differencediscretizationsof the governingflowequationsbasedon struc-tured meshes. The CFL3DEuler results that wereused in thepresent study wereobtained from Ref. 29, wherein aeroelasticmodelingsimilartothatof thepresentstudy w as implementedwithin th ecode.Steady Pressure Comparisons

Steady pressure distributionsalongt heupper surface of theN A C A 0012 airfoil are presented in Fig. 9. The calculated

steady pressures from the Euler code ar ecompared with th eexperimental steady pressure data of Ref. 30 . These pressuredistributions indicate tha t there is a moderately strong shockwavenear themidchord of the airfoil. The Euler resultscom-pare well with the experimentaldata in predicting the shocklocation and strength, as well as the overall pressure levels.This suggests that viscous effects are relatively small for thiscase.Generalized ForceComparisons

Generalizedaerodynamicforces for theN AC A0012airfpilare presented inFig.10, computedusingtheun structuredgridEuler pulse analysis, the unstructured grid Euler harmonicanalysis, and the CFL3D harmonic analysis. Both sets ofharmonic results were obtained at six values of reduced fre-quency: k = 0.0,0.125, 0.25,0.5, 0.75, and 1.0.Asshown infig. 10, the forces from the unstructured grid Euler pulseanalysisagreewellwith the forces from th eharmonicanalysis.

1.2.8.4

-Cp 0-.4-.8

-1.2

3 *Euler o E x p e r i 9o >-d ^ IIIi 111 *

i i i i i0 .2 .4 .6 .8 1.0

x/cFig. 9 Comparisonof uppersurface steadypressure distributionsonthe N A C A 0012airfoil at A/oo = 0.8 and o = 0deg.

Eule r - pulseo Eule r h a r m o n i cn CFL3D harmonic

-1.2 .4 .6 .8 1.0kFig.1 0 Comparisons of generalized aerodynamic forces computedusingCFL3Darid the cell-centered unstructured-grid Euler algorithmfor th eNACA0012 airfoil at M


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442 RAUSCH, B A T IN A , A N D Y A N G J. A I R C R A F TThe harmonicanalysis, howev er, isconsideredto be themoreaccurate of th e tw o sets of calculations because the locallinearityassumptionin thepu lse analysismay bequestionablefo r transonic flow cases. As fur ther shown in Fig. 10, theharmonicforces from the unstructured gridEulercodeare invery good agreement with th e CFL3D (Euler) harmonicforces, fo r both plunge an d pitch motions, for all values ofreduced frequency that wereconsidered.AeroelasticComparisonsAeroelasticresults forcaseAwereobtained for the NACA0012airfoil usingthe time-marchinganalysis.Thecalculations

x 10-3 CFL3D Euler

-1 0.2jfe/^ Q = 0.53|-210

-1-2

AQ = 0.8

J___J0 .04 .08 .12 .16 .20

TFig. 11 Comparisons of generalized displacements computed usingCFL3D and the unstructured-grid Euler code for the NACA 0012airfoil at A o o = 0.8 and ao= 0 deg forcase A.

Datax 10 3 Two mode fit

0.2io-13210

-1-2-3

0.8

.04 .08 .12 .16 .20 .TFig. 12 Effects of nondimensional dynamic pressure on the gen-eralizeddisplacement of thesecondcoupled modef ortheNACA 0012

airfoil at A b o = 0.8 and ad = 0 deg forcase A.

x 10 3 Mode 1 Mode2

AmplitudeQ = 0.2

-1321

Amplitude o1-2

-r

\/~~\ \_

AAI I I i

5 = 0.5

5 = 0.8I

.04 .08 .12 .16 .20TFig. 13 Effects of nondimen sional dynamic pressure on thecomponentmodesof the aeroe lastic system for theNACA0012airfoilat A/oo = 0.8 and o = 0 deg for case A .

Table 2 Comparisons between aeroelastic solutions for theN A C A 0012 airfoil at A o o = 0.8 and0 = 0 deg forcase AM od e 1 M od e 2

M et ho d0.2

0.50.8

CFL3D t ime-marchingEuler t ime-marchingCFL3D time-marchingEuler t ime-marchingCF L 3 Dt ime-marchingEuler t ime-marching

-0.011-0.0110.0040.0000.0260.017

0.7940.7900.9140.9131.0271.022

-0.091-0.068-0.185-0.148-0.173-0.223

5.3635.3535.3475.3495.2705.317

were_performed forseveralvalues of dynamicpressureinclud-ingg=0.2, 0.3, 0.4, 0.5, 0.6, 0.7,and 0.8 to obtain condi-tions that bracket th e flutter point. Figure 11 shows timeresponses of the generalized displacement of the second cou-pled mode fo r Q- 0.2, 0.5, an d 0.8, which correspond tosubcritical,near neutrallystable,and unstableaeroelasticbe-havior. Also plotted are the corresponding responses com-puted using CFL3D (Euler).29 A comparison of these re-sponses indicates that the time-marching aeroelastic resultsfrom the unstructured grid Euler code agree well with thosefrom the CFL3D code lending additional confidence in theaeroelastic modeling procedures that were implemented.Shown inFig. 12 are the two-modecurvefitsof the responsesfrom the Euler code again showing excellent approximationsto the originaldata.Th e component modes from_these curvefits are shown in Fig. 13 for the three values of Q that wereconsidered in Figs. 1 1 and 12. Similar to the flat-plate airfoilcaseof Fig. 7, the N AC A 0012airfoil results of Fig. 13showthatthe componentmodesconsist of a dom inant modecorre-sponding to bending (mode 1) and a second higher frequencymode corresponding to torsion (mode 2). Damping and fre-quency estimates from this analysis are comparedwithsimilarvaluesfrom th eCFL3Dresults inTable2.Thesecomparisonsindicatethat the unstructured gridEulervalues correlatewellwith th e CFL3D values. Also, th e flutter value fo r Q, againcomputed by quadratic interpolation of the damping values,was 0.50 for the Euler code an d 0.48 for the CFL3D code.Linear theory atM* =0.8, which of course does not includetransonic effects, predicts a much higher value of 1.89.

Concluding RemarksModifications to a two-dimensional, unsteady Euler codebased on unstructuredgridm ethods for the aeroelasticanaly-sisofairfoils weredescribed. Thisinvolved theadditionof the

structural equations of motion fo r their simultaneous timeintegrationwiththegoverningflowequations.T he flow solverof the Euler code involves a multistage, Runge-Kutta, time-stepping scheme,whichusesa finite-volumespatialdiscretiza-tion. Results for node-based and cell-centered types of dis-cretizations of the Euler equations were presented. The codealso includes a dynamic mesh algorithm that is capable oftreating general aeroelastic motions. Results were presentedfo r the flat-plate airfoiland the NAC A 0012airfoil to demon-strate applications of the Euler code to generalized aerody-namic force computation and aeroelastic analysis. Detailedcomparisons of results fromthe two-dimensional Euler code,theCFL3DEuler/Navier-Stokescode,and lineartheoryweremade.For theflat-plate airfoil,comparisonsbetweengeneral-ized forces obtained using the Euler code and linear theoryshowed that a cell-centered Euler scheme was superior inaccuracy to anode-based scheme. For the NACA 0012airfoilat transonic flow conditions, comparisons with generalizedforcesobtained from CFL3D(Euler) showedthatthecell-cen-tered unstructured grid Euler code is also accurate for tran-sonic unsteady cases.


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MAY 1990 U N S T R U C T U R E D D Y N A M I CMESHES 44 3Aeroelastic results were presented for two-degree-of-free-dom airfoils todemonstrateapplication of the Eulercode fo rsuch analyses. Results from both time-marching an d state-spaceaeroelasticanalysesusingthe Eulercodewerecomparedwithlinear theory results for theflat-plate airfoilcase.Damp-ing an d frequency characteristics of the aeroelastic modeswere ingood agreement, and the dynamic pressure at flutteragreed to within 5%. For the NACA0012airfoil, the tran-

sonic, t ime-marching, aeroelastic results compared well withsimilar results fromCFL3D and the flutter dynamic pressuresagreedto within4 .Acknowledgments

Theworkconstitutes apartof thefirstauthor'sM.S.Thesisat Purdue Universi ty and wassupportedby the NASA Lang-ley Graduate AeronauticsProgram under Grant NAG-1-372.Also,the authorswould liketo thankPareshParikhofVigyanResearchAssociates, Ham pton,Virginia,and Rainald Lohnerof George Washington Univers i ty , Washington, DC, forprovidingth egridgeneration programthatw asusedto gener-ate the grid for the NACA0012airfoil in the present study.References

'Edwards ,J. W., andTho m as ,J.L., ComputationalMethodsforUns teadyTransonicFlows, A I A APaper 87-0107,Jan. 1987.2Borland,C.J.,andRizzet ta ,D. P., NonlinearTransonic FlutterAna lysis , AIAA Journal, Vol. 20, No. 11, 1982, pp . 1606-1615.3Batina, J. T., Seidel,D. A., Bland, S. R., and Bennett, R. M., UnsteadyTransonicFlowCalculationsfor R ealisticAircraft Config-urations, AI AAPaper 87-0850,April1987.4Isogai,K., and Suetsuga, K., Numerical Simulationof TransonicFlut ter of a SupercriticalWing, National AerospaceLab., Tokyo,Japan, Rept.TR-276T,Aug. 1982.5Ide ,H.,an dShankar ,V .J., UnsteadyFullP otential A eroelast icComputations for Flexible Conf igura t ions, AIAA Paper 87-1238,J u ne1987.6Bendiksen, O. O., and Kousen , K .A., TransonicFlutter Analy-sis Using theEulerEquations, AIAAPaper 87-0911, April1987.7Kousen, K .A., an d Bendiksen, O .O., Nonlinear Aspects of theTransonic Aeroelastic Stability Problem, AIAA Paper 88-2306,Apri l 1988.8W u ,J., Kaza ,K . R.V .,an d Sankar ,L .N., A Techniquefor thePrediction of Airfoil Flutter Characteristics in Separated Flow,AIAAPaper 87-0910, April1987.9Reddy, T. S.R ., Srivastava,R .,an dKaza,K. R.V ., TheEffectsof Rotational Flow, Viscosity, Thickness, and Shape on TransonicFlutter DipPhenomena, AIAAPaper 88-2348, April 1988.10 Guruswamy,G. P., Time-AccurateUnsteadyA erodynamicandAeroelastic Calculations of Wings Using Euler Equations, AIAAPaper 88-2281,April 1988.H Mavripl i s ,D.,a ndJameson,A., Mu ltigrid Solution of the Two-Dimensional Euler Equations on Uns t ruc tu red TriangularMeshes,AIAA Paper 87-0353,Jan. 1987.

12 Morgan, K .,Peraire,J. ,Thareja,R. R., andStewart,J.R ., AnAdaptive Fini te Element Scheme for the Euler and Navier-StokesEquations, AIAAPaper 87-1172, J u ne1987.13Jameson,A., Baker, T.J., andW eatheri ll ,N. P., Calculationof InviscidTransonic Flow Over a Complete Aircraft , A I A APaper86-0103,Jan. 1986.14 Peraire, J., Peiro, J., Formaggia, L., and Morgan, K ., FiniteElement Euler Computations in Three Dimensions, AIAA Paper88-0032,Jan. 1988.15 Lohner, R., and Morgan, K ., Improved Adaptive RefinementStrategies for Finite Element Aerodynamic Computations, AIAAPaper 86-0499,Jan. 1986.16 Peraire,J.,M o rg an, K .,Peiro,J.,and Zienkiewicz,O.C., AnAdaptive Finite ElementMethod forHighSpeedFlow, AIAAPaper87-0558,Jan. 1987.17Batina,J.T., Unsteady Euler Airfoil SolutionsUsing Unstruc-tured DynamicMeshes, AIAAPaper 89-0115,Jan. 1989.18 Batina, J. T., Unsteady Euler Algorithm with UnstructuredDynamic Mesh for Complex-Aircraft Aeroelastic Analysis, AIAAPaper 89-1189, April 1989.19Anderson, W.K ., Thomas, J.L., and Rumsey, C.L., Exten-sion and Applications of Flux-Vector Splitting to Unsteady Calcula-tions on DynamicMeshes, AIAAPaper 87-1152, June1987.20Bland, S.R., Development of Low-Frequency Kernel-FunctionAerodynamics for Compar i son with Time-Dependent Finite-Differ-

enceMethod, NASA TM 83283, M ay1982.21 Seidel, D.A., Bennett , R .M ., an d Whi t low, W .,Jr., An Ex-ploratory Study of Finite Difference Grids fo r Transonic UnsteadyAerodynamics, AIAAPaper 83-0503,Jan. 1983.22Seidel, D .A., Bennett , R .M ., an d Ricketts, R .H., Some Re-cent Applications ofXTRAN3S, AIAAPaper 83-1811, July1983.23 Edwards , J .W., Bennett , R.M ., Whi t low, W .,Jr.,a nd Seidel,D .A., Time-MarchingTransonic F lutter SolutionsIncluding Angle-of-Attack Effects, Journal of Aircraft, Vol. 20, No. 11, 1984, pp .899-906.24Edwards, J. W ., Bennett, R.M., W hi tlow, W. ,Jr.,and Seidel,D .A., Time-MarchingTransonic F lutter SolutionsIncludingAngle-of-Attack Effects, AIAAPaper 82-3685, M ay1982.25Bennett , R .M., an d Desmarais, R .N ., CurveFitting ofAeroe-lastic Transient ResponseData with ExponentialFunctions, FlutterTesting Techniques, NASASP-415, M ay1975, pp .43-58.26 Batina,J .T.,a ndYang,T. Y. , Applicationof TransonicCodesto Aeroelastic Modeling ofAirfoils Including ActiveControls, Jour-na l of Aircraft, Vol. 21, No. 8, 1984, pp . 623-630.27Batina, J.T., an d Yang, T . Y ., Transonic Calculation of Air-foils Stability and Response with Active Controls, AIAA Paper84-0873, M ay1984.28Isogai, K ., Numerical Study of Transonic Flutter of a Two-Dimensional Airfoil, National Aerospace Laboratory, Tokyo,Japan, TR-617T, July 1980.29 Robinson,B .A., Time-MarchingAeroelasticAnalysiso fWingsUsing th e Euler Equat ions with a Deforming Mesh, M.S.Thesis,Purdue Univ. , West Lafayette, IN, May 1989.30 McDevit t , J. B ., an d Oku no , A. F. , Static an d Dynamic Pres-sure Measurements on a N ACA 0012Airfoil in the Ames HighReynolds NumberFacility, NASATP-2485, June1985.


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euler flutter analysis of airfoils using unstructured dynamic meshes

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