Computational-Fluid-Dynamics- And Computational
JOURNAL OF AIRCRAFTVol. 47, No. 3, May–June 2010Computational-Fluid-Dynamics- and Computational-StructuralDynamics-Based Time-Accurate Aeroelasticity ofHelicopter Rotor BladesG. P. Guruswamy NASA Ames Research Center, Moffett Field, California 94035DOI: 10.2514/1.45744A modular capability to compute dynamic aeroelastic characteristics of rotor blades using the Euler/Navier–Stokes flow equations and finite element structural equations is presented. The approach is based on a time-accurateanalysis procedure that is suitable for nonlinear fluid–structure interaction problems. Fluids and structural solversare time-accurately coupled in the C environment. Unsteady aerodynamic and aeroelastic results are validatedwith experimental data for nonrotating and rotating isolated blades.1970s, respectively. The levels of fidelity (see Fig. 1) in both areasincreased along with increased availability of computer power. Timeaccurately-coupled CFD/CSD methods for aircraft are adequatelymature for solving full configurations [11]. Many complexconfigurations have been solved using NASA’s advanced aeroelasticsoftware, HiMAP [12].In contrast to the situation for fixed-wing aircraft, the CFD/CSDbased time-accurate aeroelastic methods for rotorcraft aresignificantly lagging, as shown in Fig. 1. The arrows in Fig. 1 areapplicable to both fluids and structures. The arrow related tocomplexity indicates the ease with which complex geometries can bemodeled using lower-fidelity models. As reported in [13], the currentvalidated state of the art is the quasi-steady or loosely coupledapproach based on hybrid CFD and CC methods [14]. Computational structural dynamics used in the CC method are limited tosimple beam models, mostly in harmonic motions. Recently, hybridmethods were extended to tightly coupled computations [15], butthese have not yet been validated. Since CFD computations arecoupled in an ad hoc fashion with CC computations [15], it may notbe adequate to maintain time accuracy in nonlinear flow regimes inwhich the hybrid CFD-CC method is neither valid nor suitable fortransient conditions [13]. Furthermore, the inability to predictphase angles for flapping motion may be a strong indication ofinadequacies in the approach [16].Within the Fundamental Aeronautics Program, NASA hasinitiated an effort to develop CFD/CSD-coupled methods from firstprinciples, differing from current approaches in which CFD is run ina slave mode by hybriding with comprehensive codes [15].Development will be validated using classical test cases includingall possible physics-based modeling in a CFD/CSD-coupledapproach. From the configuration point of view, it will be more of abottoms-up approach rather than the current top-down approach[17], which tries to address missing physics such as flexibility afterthe fact. In addition, good balance in the fidelity (becomingimportant for industry [3]) will be maintained rather than the currentextreme fidelity imbalances between CFD and CSM used forrotorcraft [15]. For example, under current practices, CFD solutionsusing millions of grid points are coupled with simple 2-D beammodels [15].In this effort, beam finite element structural equations that areadequate to predict the onset of flutter are used. The primaryadvantage of beam elements is that they allow a reduction in thenumber of structural equations while maintaining accuracy inpredicting responses. In addition, due to the absence of publicdomain aeroelastic test data, they facilitate validation withcompatible comprehensive codes. Most tools developed for the2-D beam elements in the modular C environment will also beapplicable for use with 3-D finite element method (FEM)-basedCSD models to be used later in the program.IntroductionCOMPUTATIONAL aeroelasticity is one of the most challenging fields for both rotorcraft and aircraft. Flutter “thatmakes or breaks a vehicle” is an aeroelastic instability caused bystrong coupling of fluid and structural forces. In particular, thebehavior of rotary wings is inherently flexible and dynamic,requiring time-accurate methods to compute aeroelastic characteristics.Aerodynamic methods based on linear theory are well establishedfor today’s rotorcraft applications, resulting in advanced codes such asthe rotorcraft comprehensive analysis system [1] and comprehensiveanalytical model of rotorcraft aerodynamics and dynamics [2]. Thesecodes, traditionally known as comprehensive codes (CC), includestructural modeling techniques based on nonlinear beam theory. Incomprehensive codes, the linear aerodynamic parameters are eitherexpressed as a function of structural parameters or specified usingempirical theories. While comprehensive codes have performed wellin the linear flow regime, their use is stretched beyond their realm ofapplicability to nonlinear flow regimes [3]. To overcome the inabilitiesof linear theories to compute complex flows involving moving shockwaves and flow separation, methods based on the Euler/Navier–Stokes equations are essential [4]. At the same time, computationalfluid dynamics (CFD) needs to be time-accurately coupled withadvanced computational structural dynamics (CSD) capabilities tocapture the physics related to dynamic aeroelastic effects.Since development of the first unsteady 2-D transonic code,LTRAN2 [5], in the mid-1970s, use of CFD for aeroelasticity hasmade continuous impacts on both aircraft and rotorcraft. The firstvalidated results obtained by the time-accurate coupling of finitedifference-based fluids equations with finite-element-based structures equations for blade sections using LTRAN2 are reported in [6].LTRAN2 is the first CFD code applied to unsteady blade–vortexinteraction simulations [7] of rotor blades. Reviews of rotary wingaeroelasticity are reported in survey papers [8,9]. The first fullyvalidated procedure to compute CFD/CSD-based transonicaeroelasticity of fixed blades is presented in [10].Finite-element-method-based CSD and finite-difference-methodbased CFD have grown continuously since the mid-1950s and earlyPresented as Paper 4199 at the AIAA 39th Fluid Dynamics Conference,San Antonio, TX, 22–25 June 2009; received 1 June 2009; revision received18 January 2010; accepted for publication 1 February 2010. This material isdeclared a work of the U.S. Government and is not subject to copyrightprotection in the United States. Copies of this paper may be made for personalor internal use, on condition that the copier pay the 10.00 per-copy fee to theCopyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923;include the code 0021-8669/10 and 10.00 in correspondence with the CCC. Senior Scientist, Applications Branch, Advanced SupercomputingDivision; guru.p.guruswamy@nasa.gov, Associate Fellow AIAA.858
859GURUSWAMYRUNEXE can be found in [21]. Another advantage of using C isthat it is portable to other frameworks.During coupled calculations it is important to monitorconvergence data when computations are in progress. Therefore, a2-D plotting capability based on XMGRACE [22], an open-sourcesoftware module, is included in the process. In addition, it isnecessary to save data for high-end graphics visualization. In thisprocess, a capability has been added to save data using the FieldViewformat [23] at user-specified intervals. As demonstrated in [21], thisapproach is efficient for coupling nonlinear flows with nonlinearstructures required for the rotorcraft system.RUNEXE differs from HiMAP in several aspects. One differenceis that RUNEXE is independent of a data communication protocol,whereas HiMAP is hard-wired for using the message passinginterface. Both internal I/O and TCL/Tk-based [21] data communications have been tested in RUNEXE. In addition, unlike HiMAP,RUNEXE can have dynamic graphics interfaces.CFD ModuleFig. 1 Levels of fidelity used in fluid–structure interaction computations.ApproachDomain-Based ApproachWhen simulating aeroelasticity with coupled procedures, it iscommon to deal with fluid equations in the Eulerian reference systemand structural equations in the Lagrangian system. The structuralsystem is physically much stiffer than the fluid system, and numericalmatrices associated with structures are orders-of-magnitude stifferthan those associated with fluids. Therefore, it is numericallyinefficient (or even impossible) to solve both systems using a singlenumerical scheme (see the Sub-Structures section in [18]).Guruswamy and Yang [6] presented a numerical approach to solvethis problem for 2-D airfoils by independently modeling fluids usingthe finite-difference-based transonic small-perturbation equationsand structures using finite element equations. The solutions werecoupled only at the boundary interfaces between fluids and structures.The coupling of solutions at boundaries can be done either explicitlyor implicitly. This domain-based approach allows one to take fulladvantage of state-of-the-art numerical procedures for individualdisciplines. This coupling procedure has been extended to 3-Dproblems and incorporated in several advanced aeroelastic codes suchas XTRAN3S [19], based on the transonic small-perturbation theory.It was also demonstrated that the same method could be extended tomodel fluids with the Euler/Navier–Stokes equations on moving grids[10–12,20]. The coupled fluid–structure analysis procedure using adomain-based approach is described in the next section.To facilitate the domain-based approach, it is assumed that CFDand CSD solvers are independent executables. Interfaces from fluidsto structures (FTOS) and structures to fluids (STOF) are alsoconsidered separate executables. Activation of executables andcommunication among them are managed by a C executive,RUNEXE [21], and data transfers are made through I/O. Figure 2shows a flow diagram of the process. This approach provides highmodularity to the analysis process in addition to the flexibility ofusing different CFD and CSD codes. More details about C basedIn this paper, the Reynolds-averaged Navier–Stokes solverOVERFLOW (based on the diagonal form of the Beam–Warmingcentral-difference algorithm and the algebraic Baldwin–Lomaxturbulence model) with modifications to model rotor blades is usedfor flow solutions [24]. The latest version of the code OVERFLOW2is extensively validated for steady flows. In this work, validation iscarried out for unsteady flows on flexible blades. An interface hasbeen added that exports blade surface pressures to the FTOS moduleand reads-in a new deformed grid from the STOF module at everytime step.The strong conservative law form of the Navier–Stokes equationsis used for accurate modeling of nonlinear flows. The equation forpressures needed for aeroelastic equations [25] can be written asp 1 e 0:5 u2 v2 w2 where is the ratio of specific heats; e is enthalpy; is freestream airdensity; and u, v, and w are velocities nondimensionalized with thefreestream speed of air in the x, y, and z directions, respectively. Thetime variable used in the Navier–Stokes equations to obtain isnondimensionalized by the ratio of freestream airspeed and chordlength of the blade.CSD ModuleThe 10-degree-of-freedom beam finite element BEMBLD used inthis work (shown in Fig. 3) is a modified version of Bernoulli–Eulerbeam-theory-based FEM software [26]. The nodal degrees offreedom (DOF) u and w represent flapping and chordwise DOF,whereas , , and represent torsional, chordwise, and flapwiserotation DOF, respectively. It is improved by adding the first-ordereffects of centrifugal rotation [27] and torsion–bending coupling[28]. This element is well validated for nonrotating cases [26].Figure 4 shows the validation for a rotating blade [29], for which theaverage structural properties are given in Table 1. The frequency ofthe first flapping mode compare well with the shake test for rotorspeeds up to 650 rotations per minute (rpm).Element properties of BEMBLD are assembled into globalproperties [26] and incorporated in CSD based on Lagrange’sequations of motion: g fug k fug fFg m fugFig. 2Flow diagram of analysis process.(1)Fig. 3Ten degree of freedom beam finite element BEMBLD.(2)
860GURUSWAMYintegration scheme is nondissipative and does not lead to anynonphysical aeroelastic damping.The step-by-step integration procedure for obtaining theaeroelastic response is performed as follows. The grid for the flowsolver is obtained using a dynamic grid generation module from theENSAERO code [10], using assumed initial values for displacementand acceleration vectors fqg. Using this grid, thefqg, velocity fqg,aerodynamic force vector fFg at time t t is computed fromOVERFLOW2. Based on this aerodynamic vector, the newdisplacements at time t t are computed by solving Eq. (3). Thisprocess is repeated every step to advance the aerodynamic andstructural equations of motion forward in time until the requiredresponse is obtained.Fig. 4 Effect of rotor speed on the flapping mode frequencies.where m , g , and k are mass, damping, and stiffness matrices,respectively. fFg is the aerodynamic force vector defined as1 2 U2 fLg, where fLg is the aerodynamic global nodal force vector, is the freestream density, and U is the local speed of the bladesection.The aeroelastic equations of motion (2) are solved by a numericalintegration method based on the linear-acceleration method [30].Assuming a linear variation of the acceleration, then velocities anddisplacements at the end of a time interval t t can be derived asfollows: t tt t fugt fugt fugt t(3a)fug22 t fugt t fugt t fug 2 2 t t t t t (3b)fugfug36fugt t D fFgt t G fvg K fwg (3c)where 2 1 t t G K D M 26 t tt fugv fug2 t tt t fug t fugw fugt t fug2These time integration equations can also be derived by using thesecond-order time-accurate central-difference scheme, which fallsinto the explicit form of Newmark’s time integration methods [30].To obtain physically meaningful responses, it is necessary to use thesame time step for integration for both the fluids and aeroelasticequations of motion. Though Eqs. (3) are explicit in time, the timestep size required to solve Reynolds-averaged Navier–Stokesequations is an order of magnitude less than that required to solve theaeroelastic equations of motion (3). In addition, the above timeTable 1ResultsComputations are made for isolated nonrotating and rotatingblades. A C-H grid topology with 151 points in the chordwisedirection, 45 grid points in the spanwise direction, and 50 points inthe normal direction is used.Nonrotating BladesFlow computations are made using the OVERFLOW2 code alongwith the Baldwin–Lomax turbulence model [24]. Accurateprediction of unsteady pressures is a necessary part of aeroelasticcomputations, and the current unsteady results are validated withwell-documented experimental results reported in [31] for anonrotating blade. In the experiment, a blade with an aspect ratio of 6and a 6% circular arc section is subjected to forced sinusoidal elasticbending motion, and corresponding unsteady pressure data ismeasured. Experimental data include steady-state measurementswhen the blade is not oscillating.First, steady-state computations for a nonoscillating case are madeto check the adequacy of the grid used. Good comparison betweencomputed and measured steady-state data in Fig. 5 demonstrates thatthe C-H grid size selected is adequate for resolving the transonicflows.Figure 6 shows the comparison of unsteady pressure betweenOVERFLOW2 and experiments [31] for M/ 0:90 at a reducedfrequency k 0:26 based on the chord when the blade is undergoingforced sinusoidal elastic bending motion. Both the magnitude andphase angles of unsteady pressures peak near the shock wave, whichare accurately predicted by OVERFLOW2 with 1200 time steps percycle. It is noted that the linear aerodynamics theory used in thecomprehensive code cannot predict unsteady pressure jumps andphase angles associated with moving shock waves. Flapping motionsplay an important role in the aeroelasticity of rotorcraft.A nonrotating blade for which a measured pitch-flap flutterboundary is given in [32] is selected to validate the C executiveRUNEXE. Figure 7 shows the first bending and torsional modeshapes and frequencies computed from BEMBLD. Figure 8 showsthe stable, unstable, and neutrally stable responses of the elastic blade.These responses were computed using a nondimensional time stepof 0.01, based on accuracy limits of the flow solver, which is 150times smaller than the smallest time step required to resolve the firstbending mode. This justifies the use of explicit time integrationStructural properties of NASA LangleyResearch Center bladePropertiesLength LChord cMass per unit length mBending modulus of elasticity ETorisional modulus of elasticity JPoisson’s ratio Sectional area ATorsional area moment of inertia JFlapwise area moment of inertia IxChordwise area moment inertia IyValues54.25 in.4.24 in.0:039 lb in:1:0 107 lb in:23:7 107 lb in:20.300:224 in:20:00485 in:40:00151 in:40:02447 in:4Fig. 5 Comparison of steady pressures at M/ 0:90, 0:0 deg,and Re 4:5 106 .
GURUSWAMY861Fig. 9 Comparison of upper surface pressure coefficients for rotatingblade.selected in this work. Dynamic pressure at the neutrally stablecondition for Mach 0.715 agrees well with the experiment [32].Rigid Rotating BladeFig. 6 Unsteady pressures for flapping motion at M1 0:90 andk 0:26.Results in Figs. 5–8 establish that aeroelastic computations areaccurate when running coupled computations for nonrotating blades.Next, results are demonstrated for a rotating blade, which has anaspect ratio of 6 and a NACA0012 airfoil section [33]. Figure 9shows the upper surface pressures at the 60% span station when theblade is rotating at 1500 rpm with a zero collective angle of attack inhover. Computed surface pressures converged to a steady state afterthree rotations and compare well with the experiment.Figure 10 shows comparison of Euler and Navier–Stokesequations-based sectional lift coefficients with measured data for thesame blade at 1250 rpm with a collective angle of attack of 8 deg.Results from the Navier–Stokes equations compare better than thosefrom the Euler equations beyond 70% span. Results from the liftingline theory [33] are closer to the Euler solutions beyond 60% span butdeviate away as the span station gets closer to the root. Bothcomputed results are within the uncertainty bound stated in the reporton experiment [33] for span stations beyond the 60% station.Flexible Rotating BladeFig. 7 First two modes of the blade; mode 1 is bending and mode 2 istorsion.Fig. 8 Dynamic aeroelastic responses at M 0:715 corresponding tomeasured flutter dynamic pressure q 1:31 psi, from [32].Aeroelasticity of rotating blades is not as well understood as that offixed wings. Quite often, it does not appear explicitly, since it isembedded in stability computations and implicitly controlled bytrim. With fixed wings, aeroelastic phenomena occur due to theexchange of energy between fluids and structures [34], and phaseangles between structural motions and aerodynamic responses play akey role in aeroelasticity [35]. Most studies in aeroelasticity usingcompressive codes are combined with trim conditions selected basedon wind-tunnel data criteria [36]. Independent study of aeroelasticityis required for more complex modern rotor blade configurations.Nonlinear effects of the flow also play a significant role in rotatingwing aeroelasticity, particularly when tip Mach numbers reach thetransonic range. As shown in experiments [31], the phase angle takesa jump near the shock wave, which can further impact aeroelasticbehavior.Fig. 10 Comparison of sectional lift coefficient at 1250 rpm and 8 degcollective angle of attack.
862GURUSWAMYTable 2Structural properties of MassachusettsInstitute of Technology bladePropertiesLength LChord cMass per unit length mAxial rigidity EATorsional rigidity GJBending rigidity about x axis EIxBending rigidi
Computational-Fluid-Dynamics- and Computational-Structural-Dynamics-Based Time-Accurate Aeroelasticity of Helicopter Rotor Blades G. P. Guruswamy NASA Ames Research Center, Moffett Field, California 94035 DOI: 10.2514/1.45744 A modular capability to compute dynamic aeroelasti
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