Unsteady Aerodynamic And Aeroelastic Calculations For .
VOL. 28, NO. 3, MARCH 1990AIAA JOURNAL461Unsteady Aerodynamic and AeroelasticCalculations for Wings Using Euler EquationsGuru P. Guruswamy*NASA Ames Research Center, Moffett Field, CaliforniaA procedure to solve simultaneously the Euler flow equations and modal structural equations of motion ispresented for computing aeroelastic responses of wings. The Euler flow equations are solved by a finite-difference scheme with dynamic grids. The coupled aeroelastic equations of motion are solved using the linear-acceleration method. The aeroelastic configuration adaptive dynamic grids are time-accurately generated using theaeroelastically deformed shape of the wing. The unsteady flow calculations are validated with the experiment,both for a semi-infinite wing and a wall-mounted cantilever rectangular wing. Aeroelastic responses are computed for a rectangular wing using the modal data generated by the finite-element method. The robustness ofthe present approach in computing unsteady flows and aeroelastic responses that are beyond the capability ofearlier approaches using potential equations are demonstrated.IntroductionIN the last two decades, there have been extensive developments in computational aerodynamics that constitute a major part of the general area of computational fluid dynamics(CFD).1'2 Such developments are essential to advance the understanding of the physics of complex flows, complement expensive wind-tunnel tests, and reduce the overall design cost ofan aircraft. The CFD capabilities have advanced closely inphase with the improvements in computer resources. In general, computational aerodynamics can be classified based onthe type of equations used and complexity of the configurationconsidered. The problem can be further grouped into thoseinvolving steady and unsteady flows.Computational methods to analyze wing problems involvingsteady flows have advanced to the level of using the NavierStokes equations, and calculations on wing-body configurations are in progress.3 In comparison, for unsteady flows associated with moving components such as oscillating wings, themost advanced codes use methods based on the potential-flowtheory.4 Some of the reasons for the lag in the development ofunsteady methods when compared to steady methods are 1) thecomplexity of physics associated with the movement of flexiblecomponents, 2) the complexity in modeling the flow because ofthe moving grids, and 3) the lack of development of fast,time-accurate methods. One of the main reasons to developefficient unsteady methods is to understand the interaction ofthe flows with moving structural components such as theaeroelasticity of an aircraft.Aeroelasticity plays an important role in the design anddevelopment of an aircraft, particularly modern aircraft,which tend to be flexible for high maneuverability. Severalphenomena that can be dangerous and limit the performanceof an aircraft occur because of the interaction of the flow withPresented as Paper 88-2281 at the AIAA/ASME/ASCE/AHS 29thStructures, Structural Dynamics, and Materials Conference, Williamsburg, VA, April 18-20, 1988; received May 10, 1988; revision receivedJan. 6, 1989. Copyright 1989 American Institute of Aeronauticsand Astronautics, Inc. No copyright is asserted in the United Statesunder Title 17, U.S. Code. The U.S. Government has a royalty-freelicense to exercise all rights under the copyright claimed herein forGovernmental purposes. All other rights are reserved by the copyrightowner.* Research Scientist, Applied Computational Fluids Branch. Associate Fellow AIAA.flexible components. For example, aircraft with highly sweptwings experience vortex-induced aeroelastic oscillations.5'6Also, several undesirable aeroelastic phenomena occur in thetransonic range that are due to the presence and movement ofshock waves. Limited wind-tunnel and flight tests have shownaeroelastically critical phenomena such as a low transonic flutter speed. Because of the high cost and risk involved, it is notpractical to conduct extensive aeroelastic tests. An aeroelasticwind-tunnel experiment is an order of magnitude more expensive than a parallel experiment involving only aerodynamics.By complementing the wind-tunnel experiments with the computational techniques, the overall cost of the development ofan aircraft can be considerably reduced.At present, the most advanced codes used for aeroelasticanalyses, such as XTRAN3S,7 use the transonic small-perturbation equation. Currently, ATRAN3S, the NASA-Ames version of XTRAN3S, is being used for generic research in unsteady aerodynamics and aeroelasticity of full-span, wingbody configurations.4 Although codes based on the potentialflow theory give some useful results, they cannot be used forcases involving complex flows. Now, given the availability ofnew efficient numerical techniques and faster computers,8 it istime to consider Euler/Navier-Stokes equations for aeroelasticapplications.Codes based on the Euler/Navier-Stokes equations have already been applied for practical and interesting problems involving steady flows. Generic codes, such as ARC3D,9 NASAAmes Research Center's three-dimensional Euler/NavierStokes code, have been used for several scientific investigations. Such generic codes have resulted in useful codes such asTNS, N AS A- Ames's Transonic Navier-Stokes code based onzonal grids. TNS has successfully computed complex separated flows on wings and wing-body configurations.3In this paper, a computational method for computing unsteady flows and aeroelastic responses of flexible wings is presented. This work is a part of a larger effort within the AppliedComputational Fluids Branch of NASA Ames Research Center to develop a new code, ENS AERO, an efficient generalpurpose code to compute unsteady aerodynamics and aeroelasticity of aircraft using the Euler/Navier-Stokes equations.This new code is being designed in a modular fashion to adaptseveral different numerical schemes suitable for accurateaeroelastic computations. The candidate flow solvers arebased on schemes such as central-difference schemes with artificial viscosity,10 upwind schemes,11'12 etc., which have mostly
462G. P. GURUSWAMYbeen applied for steady flows and, in some cases, for unsteadyflows over stationary bodies. The basic coding of ENSAEROcan accommodate zonal grid techniques for efficient modelingof full aircraft.3 ENSAERO is also being designed in such away that instead of the modal equations of motion, finiteelements can be directly used for more complete modeling ofstructures.This paper describes the first version, ENSAERO version1.0, which is based on the Euler equations coupled with themodal structural equations of motion. Although computationsin this paper are done using the Euler equations coupled withthe modal structural equations of motion, all techniques developed in this work can be easily extended for computationsusing the Navier-Stokes equations directly coupled with thestructural equations based on the finite-element method. Theprocedure to solve the flow equations in ENSAERO version1.0 is based on the diagonal algorithm form13 of the BeamWarming central-difference scheme.10 For the first time, thisfinite-difference scheme is adopted for aeroelastic computations using configuration adaptive dynamic grids. The coupledflow and modal structural equations of motion are solvedusing a simultaneous-integration technique based on the linear-acceleration technique. Dynamic grids that are adaptive tothe deforming shapes of the wing are generated using an algebraic grid-generation scheme. The unsteady flow computations are validated with the experimental data for both a semiinfinite wing oscillating in pitching mode, and a wall-mountedcantilever finite wing oscillating in the first bending mode. Theaeroelastic responses are computed for a rectangular wing using the modal data generated by the finite-element method andthey are correlated with the experiment. The robustness of thepresent approach in computing aeroelastic responses that arebeyond the capability of earlier approaches using potentialflow equations is demonstrated.Governing Aerodynamics Equationsand ApproximationsAIAA JOURNALTo enhance numerical accuracy and efficiency and to handleboundary conditions more easily, the governing equations aretransformed from the Cartesian coordinates to general curvilinear coordinates usingT ti? i\(x,y,z,t)(4)The resulting transformed equations can be written in nondimensional form as(5)where indicate the transformed quantities, and wherepUpuU iPpupvpwepwUPWpV, G ,(6)(e p)V- ntpand 7a)The strong conservation law form of the Euler equations isused for shock-capturing purposes. The equations in Cartesiancoordinates in nondimensional form14 can be written asV lit rixu riyV rizw(7b)(7c)dtdxdy(1)BzIn Eqs. (7), t/, V, and W are contravariant velocity components written without metric normalization. Here, the Cartesian derivatives are expanded in , r;, f space via chain-rulerelations such aswherepQ pupvpwepupu2 ppuvE puw(e p)upvpuvF pv2 ppvw(e p)v(8)Finally, the metric terms are obtained from chain-rule expansion of * , y etc., and solved for x , %y, etc., to give(9a)pw(9b)G pvwpw2 p(e p)w(2)(9d)The Cartesian velocity components u, v, and w are nondimensionalized by a (the freestream speed of sound); density p isnondimensionalized by pw\ the total energy per unit volume eis nondimensionalized by POO# , and the time / is nondimensionalized by c/a, where c is the root chord. Pressure can befound from the ideal gas law asp (y-l)[e-0.5p(u2 v2 w2)]and throughout 7 is the ratio of the specific heats.(9c)(9e)(90and(10)(3)Several numerical schemes have been developed to solve Eq.(5). In this work, the algorithm developed by Beam and Warm-
MARCH 1990UNSTEADY AERODYNAMIC AND AEROELASTIC WINGSing10 and the diagonal algorithm extension reported by Pulliam and Chaussee,13 both based on implicit approximate factorization, are used. Both algorithms were implemented in anew code, ENS AERO, a general-purpose aeroelastic codebased on the Euler/Navier-Stokes equations and the modalstructural equations of motion with time-accurate aeroelasticconfiguration adaptive grids. (Results presented in this paperare from ENS AERO version 1.0, which uses the diagonal algorithm to solve the Euler equations.)The diagonal algorithm used in this paper is a simplifiedversion of the Beam-Warming scheme. In the diagonal algorithm, the flux Jacobians are diagonalized so that the computational operation count is reduced by 50%. The diagonalscheme is first-order accurate in time and yields time-accurateshock calculations in a nonconservative mode. The schemeused is briefly explained in the Appendix. More details of thisscheme can be found in Ref. 13.Aeroelastic Equations of MotionThe governing aeroelastic equations of motion of a flexiblewing are solved by using the Rayleigh-Ritz method. In thismethod, the resulting aeroelastic displacements at any time areexpressed as a function of a finite set of assumed modes. Thecontribution of each assumed mode to the total motion is derived by Lagrange's equation. Furthermore, it is assumed thatthe deformation of the continuous wing structure can berepresented by deflections at a set of discrete points. Thisassumption facilitates the use of discrete structural data, suchas the modal vector, the modal stiffness matrix, and the modalmass matrix. These can be generated from a finite-elementanalysis or from experimental influence coefficient measurements. In this study, the finite-element method is employed toobtain the modal data.It is assumed that the deformed shape of the wing can berepresented by a set of discrete displacements at selectednodes. From the modal analysis, the displacement vector (d}can be expressed as(11)where [i/O is the modal matrix and {q } is the generalized displacement vector. The final matrix form of the aeroelasticequations of motion is(12) [F]where [M], [G], and [K] are modal mass, damping, and stiffness matrices, respectively. The [F] is the aerodynamic forcevector defined as ( V 2 ) p U l [ f l T [ A ] { ACp ) and [A] is the diagonal area matrix of the aerodynamic control points.The aeroelastic equation of motion (12) is solved by a numerical integration technique based on the linear-accelerationmethod.15 Assuming a linear variation of the acceleration, thevelocities and displacements at the end of a time interval t canbe derived as follows:( q ] t (q]t-* {?},-A, }t lD]((F\t-lG][v}-lK](w}) )'(13c)463wherev /-A/ These equations can also be derived by using the second-order-accurate central-difference scheme. Since Eqs. (13) are explicit in time, the computational time-step size is restricted bystability considerations. However, the time-step size requiredto solve the aerodynamic equation [Eq. (5)] accurately is always far less than the time-step size required to obtain stableand accurate solution using the preceding numerical-integration scheme.15 To obtain physically meaningful time-accuratesolutions, it is necessary to use the same time-step size inintegrating Eq. (5) and Eq. (12). Also, the preceding scheme isnumerically nondissipative and does not lead to any nonphysical aeroelastic phenomenon.The step-by-step integration procedure for obtaining theaeroelastic response was performed as follows. Assuming thatfreestream conditions and wing-surface boundary conditionswere obtained from a set of selected starting values of thegeneralized displacement, velocity, and acceleration vectors,the generalized aerodynamic force vector (F(t)} at time t wascomputed by solving Eq. (5). Using this aerodynamic vector,the generalized displacement, velocity, and acceleration vectors for the time level t are calculated by Eq. (12). From thegeneralized coordinates computed at the time level t , the newboundary conditions on the surface of the wing are computed.With these new boundary conditions, the aerodynamic vector{ F(t) } at the next time level, t At , is computed using Eq. (5).This process is repeated every time step to advance the aerodynamic and structural equations of motion forward in time untilthe required response is obtained.Aeroelastic Configuration Adaptive GridsOne of the major complexities in computational aerodynamics using the Euler equations lies in the area of grid generation. For the case of steady flows, advanced techniques suchas zonal grids3 are currently being used. Grid-generation techniques for aeroelastic calculations that involve moving components are in an early stage of development. The effects of theaeroelastic configuration adaptive dynamic grids on the stability and accuracy of the numerical schemes are yet to be studiedin detail.This work developed an analytical grid-generation techniquefor aeroelastic applications. This scheme satisfies the generalrequirements of a grid required for implicit finite-differenceschemes used in the present analysis.10'13 Some of the requirements are 1) grid lines intersect normal to the wing surface inthe chordwise direction, 2) a smooth stretching of grid cellsaway from the wing surface, 3) outer boundaries located faraway from the wing to minimize the effect of boundary reflections, and 4) a grid that adapts to the deformed wing positionat each time step. The type of grid used is C-H grid. The gridis generated at every time step based on the aeroelastic positionof the wing computed using Eq. (12) as follows.At the end of every time step, the deformed shape of thewing is computed using Eq. (11). The and 77 grid distributionson the grid surface corresponding to the wing surface (f gridindex 1) are obtained from previously assumed distributions. These distributions are selected to satisfy the generalrequirements of a grid for accurate computations. In thiswork, the grid in the direction is selected such that the gridspacing is fine on the wing and it stretches exponentially toouter boundaries. The grid near the nose is finer than the rest
464G. P. GURUSWAMYof the wing in order to model the nose geometry accurately. Inthe spanwise direction, a uniformly distributed grid spacing isused on the wing. In order to model the wing tip, a finer gridspacing is used near the wing tip. Away from the wing tip, therj grid spacing stretches exponentially. The f grid spacing iscomputed every time step using the deformed shape of thewing computed by using Eq. (11). The f grid lines start normalto the surface in the chordwise directions and their spacingstretches exponentially to a fixed outer boundary/In order torestrain the outer boundaries from moving, the grid is shearedin the f direction. The metrics required in the computationaldomain are computed using the following relations:& -x x - y y - z z(14a)Vt - —xrrjx — JMy — Zrfz(14b)tt -XT{x-yTty-ZT[z(14C)The grid velocities xT9 yT, and ZT required in Eqs. (14) arecomputed using the grids at new and old time levels. Thisconfiguration adaptive grid-generation scheme is incorporatedin ENSAERO. Figures la and Ib show grids around a wingsection in a x-z plane at 50% span for two positions of arectangular wing. Similarly, Fig. 2a and 2b show grids in a x-zplane at the 50% chord axis for two positions of a rectangularwing in bending motion. Both Figs. 1 and 2 show the ability ofthe grid to conform to the deforming wing surface.ResultsAIAA JOURNALthe present flow computations. The unsteady results shown arefrom the computations made for wings undergoing prescribedforced motions.Time accuracy is an essential requirement for aeroelasticcomputations. Numerical schemes used for flow calculationsin aeroelasticity must guarantee the correct calculation of amplitude and phase of unsteady pressures. In order to verify thetime accuracy of the flow calculations, a semi-infinite wingwith an NACA 64AGIO airfoil section was selected. This casewas selected to assess the accuracy of the computed resultsagainst the available two-dimensional experimental data.16For the semi-infinite rectangular wing configuration considered, a C-H type grid with 110 points in the chordwise direction, 20 points in the spanwise direction, and 20 points in thenormal direction, was used. Using this dynamic grid whichadapts according to the airfoil position as shown in Fig. 1,several unsteady calculations were made for the semi-infinitewing. All computations were made at a transonic Mach number of 0.8 for a reduced frequency of 0.1 (based on the fullchord) when the wing was oscillating sinusoidally in pitchmode with an amplitude of 1.03 deg about a mean angle ofattack of -0.21 deg.In order to study the accuracy and stability of the scheme,inviscid computations using the Euler equation [Eq. (5)] weremade with 720, 1080, and 1440 time steps/cycle. The computations were started with freestream flow conditions as initialdata, and continued for three cycles of wing oscillation toobtain a periodic solution. A Fourier analysis was carried outon the third cycle of the solution to obtain real and imaginaryAerodynamic Computations for a Semi-infinite WingIn this section, steady and unsteady computations are madeto validate the accuracy and to demonstrate the capability ofa)b)Fig. 1 Chordwise view of grid in x-z plane at 50% semispan location:a) initial position, and b) deformed position.Fig. 2 Spanwise view of grid in y-z plane at 50% chord location:a) initial position, and b) deformed position.
MARCH 1990UNSTEADY AERODYNAMIC AND AEROELASTIC WINGSNACA64A010SEMI-INFINITE WING30rM 0.796O- ——— TSP, ATRAN2M 0.796——— 1080 AND 1440 TIMESTEPS/CYCLE20-EULERNACA 64A010SEMI-INFINITE WING30EULER CALCULATIONS-- -- 720 STEPS/CYCLEk 1221)k 201010-1020 r-1020 r1010CJO-10.4.61.0x/c-10.4.61.0Fig. 4 Comparison of unsteady pressures between Euler and potential flow calculations.x/cFig. 3 Real and imaginary components of unsteady lower surfacepressures at midspan of the semi-infinite wing.NACA 64A010SEMI-INFINITE WING50M 0.7964
steady aerodynamics and aeroelasticity of full-span, wing-body configurations.4 Although codes based on the potential-flow theory give some useful results, they cannot be used for . purpose code to compute unsteady aerodynamics and aeroe-lasticity of aircraft using the Euler/Navier-Stokes equations.File Size: 1MB
Figure 1: Coupling of linear structural model and nonlinear unsteady aerodynamics within an aeroelastic CFD code such as FUN3D. A CFD-based aeroelastic system (such as the FUN3D code) consists of the coupling of a nonlinear unsteady aerodynamic system (ow solver)
Vibration theory. Steady and unsteady flows. Mathematical foundations of aeroelasticity, static and dynamic aeroelasticity. Linear unsteady aerodynamics, non-steady aerodynamics of lifting surfaces. Stall flutter. Aeroelastic problems in civil engineering structures. Aeroelastic problems of
developed to compute unsteady transonic aerodynamics for aeroelastic applications. In particular, codes that solve the small-disturbance potential equations for transonic flows about oscillating airfoils, such as LTRAN2,3 are now used routinely.4'5 Similar codes are now being developed for the computation of
The Aerodynamic Analysis and Aeroelastic Tailoring of a Forward-Swept Wing. (Under the direction of Dr. Charles E. Hall, Jr.) The use of forward-swept wings has aerodynamic benefits at high angles of attack and in supersonic regimes. These consist of reduction in wave drag, profile drag, and increased high angle of attack handling qualities.
on aerodynamic performance. The effect on the aerodynamic behaviour of the seam on the garments has not been well studied. The seam on sports garments plays a vital role in aerodynamic drag and lift. Therefore, a thorou gh study on seam should be undertaken in order to utilize its aerodynamic advantages and minimise its negative impact.
5.1.1 Unsteady Aerodynamics of Flapping Flight 136 5.1.2 Flexible wings and Aeroelasticity analysis 138 5.2 Methodology 140 5.2.1 Flow solver 140 5.2.2 Elastic Membrane Model 143 5.2.3 Coupling 143 5.3 Results 144 5.3.1 Wing Configuration and K
There are many phenomena associated with aeroelasticity that challenge today’ s aeroelastic analysis methods, particularly the components of these analyses simulating vehicle aerodynamics. At transonic speeds steady and unsteady aerodynamic effects tend to reduce the ‘ utter dynam
Academic writing is a formal style of writing and is generally written in a more objective way, focussing on facts and not unduly influenced by personal opinions. It is used to meet the assessment requirements for a qualification; the publ ication requirements for academic literature such as books and journals; and documents prepared for conference presentations. Academic writing is structured .
