Computational Aeroelasticity of Rotating Wings with Deformable

Airfoils

Smith Thepvongs James R. CookGraduate Research Assistant Graduate Research Assistant

sthepvon@umich.edu jcook30@mail.gatech.edu

Carlos E.S. Cesnik Marilyn J. SmithProfessor Associate Professor

cesnik@umich.edu marilyn.smith@ae.gatech.edu

Department of Aerospace Engineering School of Aerospace EngineeringUniversity of Michigan, Ann Arbor, MI Georgia Institute of Technology, Atlanta, GA

Abstract

This paper presents a simulation for high-fidelity aeroelastic analysis of rotating wings with camber-wisestructural flexibility and embedded actuators. An unstructured Reynolds-Averaged Navier-Stokes (RANS)computational fluid dynamics (CFD) solver is coupled with a non-linear structural dynamics analysis. TheCFD solution uses overset grids to combine the stationary and moving frames of reference. The structural for-mulation expands the conventional one-dimensional beam representation with additional degrees-of-freedomto capture plate-like cross-sectional deformations while allowing an arbitrary distribution of active and pas-sive materials in the cross section. Motion and forces on the non-coincident fluid and structural grids aretransferred using a finite-element-based interpolation, along with a least-squares fit for extrapolations. Trimand convergence to periodic response are assisted by a low-order analysis that is also discussed. Finally,as an initial verification of the implementations, results from the low-order and CFD-based solutions arecompared for a rigid-airfoil rotor in forward flight.

Introduction

Reducing noise, vibration and power consumptionis becoming increasingly important to the modernhelicopter industry. This has inspired the develop-ment of many new technologies. Especially promis-ing are those based on active blade control whichhave a potential ability to adapt to flight conditionsas well as improve multiple characteristics throughthe same system. Such concepts have tradition-ally achieved control without intentionally deform-ing the airfoil’s main supporting structure. Recently

Presented at the American Helicopter Society65th Annual Forum, Grapevine, Texas, May 27-29,2009. Copyright c©2009 by the American HelicopterSociety, Inc. All rights reserved.

however, actuation systems have been proposed thatapply continuous shape changes through the use ofintegrated active materials [1, 2]. These morphing-type systems may offer several benefits, includingimproved aerodynamic efficiency as well as poten-tial for good structural weight efficiency if the actu-ators themselves contribute to blade overall stiffnessand strength. Although some aspects of these sys-tems have been previously studied, general designrequirements and effectiveness are still largely un-known.

In order to analyze morphing-type rotors indepth, one must consider several phenomena nor-mally assumed to be unimportant in rotor problems.Structurally, shape changes can be introduced by acombination of cross-section flexibility and actuatorforces. Bending along the chordline cannot be mod-eled in the beam formulations typically used for ro-tor analyses, while the coupled actuator/structural

dynamics may be difficult to analyze when the actu-ator is highly integrated. Aerodynamically, the pos-sibility of general conformations adds a large degreeof complexity. Besides altering the behavior associ-ated with linear phenomena, shape changes can alsosignficantly change the effects of compressibility orviscosity. To properly assess the impact on vibra-tion and power, these higher-order effects must beconsidered.

In the modern aeroelastic analysis of rotors, non-deforming cross sections are generally assumed inthe structural and aerodynamics theories, as well asin their coupling. The structural theories are basedon classical beam degrees-of-freedom, and the aero-dynamics use either a lifting line model (often aug-mented with semi-empirical models to account forhigher-order effects), or computational fluid dynam-ics. For the latter, the current state-of-the-art isReynolds-Averaged Navier Stokes (RANS). Exam-ples of the successful coupling of structural dynam-ics analyses with RANS-CFD include CAMRAD-OVERFLOW [3], DYMORE-OVERFLOW [4],RCAS-OVERFLOW [5], UMARC-TURNS [6], andHOST-elsA [7]. Though significant progress hasbeen made in CFD-based aeroelastic simulation, theability to model deformable airfoils (with or withoutactuation) while also having sufficient accuracy fornoise, vibration and power assessment has not beendeveloped to the authors’ knowledge.

This paper describes the development of a high-fidelity simulation for the aeroelastic analysis of ro-tors with actuated, flexible airfoils. The simulationconsists of a RANS CFD analysis coupled three-dimensionally with a structural analysis that cap-tures camber deformations. The simulation is sup-ported by a finite-state, semi-empirical aerodynam-ics analysis, which is used to assist initialization andtrim. The theories and coupling strategies of thedifferent formulations are presented. A preliminarycomparison of the aeroelastic response predictionsfrom the low-order and CFD-based analyses is alsoshown for a rotor in forward flight.

Structural Dynamics Formula-tion

The computational structurual dynamics (CSD)formulation used in the current study has been pre-sented in Refs. 8 and 9. It follows the variational-asymptotic method for the analysis of compos-ite beams [10]; that is, the equations of motion

for a slender anisotropic elastic three-dimensionalsolid are approximated by the recursive solutionof a linear two-dimensional problem at each crosssection [9], and a one-dimensional geometrically-nonlinear problem along the reference line [8].This procedure allows the asymptotic approxima-tion of the three-dimensional warping field in thebeam cross sections, which are used with theone-dimensional beam solution to recover a three-dimensional displacement field. The present imple-mentation adds an arbitrary expansion of the dis-placement field through a set of functions approx-imating the sectional deformation field to capture“non-classical” deformations, which are referred toas finite-section modes.

The geometrically-nonlinear dynamic equationsof equilibrium along a reference line of a slendersolid were presented in Ref. 8. If all magnitudes areexpressed in their components in a reference frameattached to the deformed reference line, they arewritten as:

(d

dt+ ΩB)PB = (

d

dx+ KB)(FB − f1) + f0

(d

dt+ ΩB)HB + VBPB =

(d

dx+ KB)(MB −m1) + (e1 + γ)FB + m0

d

dtQt =

d

dx(Qs1 − fs1 − (Qs0 − fs0)

(1)

The first two sets of equations imply equilibriumof forces and moments, where K is the current cur-vature of the reference line, the pairs (V, Ω) and(P,H) are the local translational and rotational in-ertial velocities and momenta, respectively, and Fand M are the sectional forces and moments (similarto traditional geometric-nonlinear blade modelingin state-of-the-art rotorcraft aeromechanics codes).In addition, f0 and m0 are the conventional (zero-order) applied forces and moments, respectively, perunit length on the beam, while f1 and m1 are thefirst-order loads associated to the work needed todeform the cross section. The last set of equa-tions includes the equilibrium of the generalizedforces (Qs0 , Qs1) and momenta, Qt, correspond-ing to the finite-section modes, which are definedby prescribed cross-sectional displacement shapesψq(x2, x3) with amplitude q(x). These equations arecomplemented by the cross-sectional constitutive re-lations, which, for a elastic solid with embedded ac-

tuation, are derived in Ref. 8 using the variational-asymptotic method. These equations take the finalform of

FB

MB

Qs0

Qs1

= [S]

γκqq′

−

F (a)

M (a)

Q(a)s0

Q(a)s1

PB

HB

Qt

= [M ]

VB

ΩB

q

(2)

where [S] and [M ] are the cross-sectional stiffnessand mass matrices, respectively. The superscript(a) indicates actuation loads per unit length on thestructure, and γ and κ are the beam strain measuresassociated to forces and moments, respectively. Thevariational asymptotic method determines [S], [M ]and actuation loads by discretizing the cross sectioninto finite elements, allowing an arbitrary geometryand material distribution.

To develop a finite-element solution, the dynam-ics of the member are first described by a mixedformulation in which the variation of the equationsof virtual work are taken with regard to displace-ment, force and momentum variables. These vari-ables are used to define the structural state. Strain-displacement relations and velocity-displacement re-lations are satisfied simultaneously by using La-grange multipliers. The resulting governing equa-tions are first-order in time and space, which allowsthe use of simple integration schemes.

The solution of these equations has been imple-mented in the computer code UM/NLABS (Univer-sity of Michigan’s Nonlinear Active Beam Solver).

Low-Order Aerodynamics For-mulation

Deforming Thin-Airfoil Theory

The low-order model uses the two-dimensionalfinite-state formulation for flexible airfoils presentedin Ref. 11. It is based on a Glauert expansion ofthe potential flow equations for a deformable airfoilof infinitesimal thickness. The camber-wise airfoildeformation is written as:

h(ξ) =∞∑

n=0

hnTn(ξ)

Ln = −b

∫ 1

−1

Tn(ξ)∆Pdξ (3)

where h is the local displacement normal to thechord, b is the semi-chord length and ξ is the non-dimensional coordinate along the chordwise direc-tion. Tn are the Chebyshev polynomials of the firstkind, which define the generalized camber-wise dis-placement amplitudes, hn , and its associated air-loads, Ln integrated from the local airloads ∆P .The final result (Ref. 11) for the airloads computedin the normal and chord directions arising from thepotential flow assumptions is:

1ρc`α

L = −b2M(¨h + ˙ν)

−bu0C( ˙h + ν − λ0)− u20Kh

−bG(u0h− u0ν − u0λ0) (4)1

ρc`α

D0 = −b( ˙h + ν − λ0)T S( ˙h + ν − λ0)

+b(¨h + ˙ν)T Gh

−u0( ˙h + ν − λ0)T (K −H)h+(u0h− u0ν + u0λ0)T Hh (5)

L = L0, L1, Ln, ...T

ν = ν0, ν1, 0, ...T

λ0 = λ0, 0, 0, ...T (6)

where L is the vector containing the generalized air-loads, and similar definitions are given for the gener-alized camber-wise displacements (h), rigid-sectionnormal velocity (ν), and section inflow distribution(λ). The drag force is D0, and M,S, G, K, andH are constant coefficient matrices whose completedefinition is given in Ref. 11.

Dynamic Wake

The wake-induced velocity (λ0 in eqn. 6) is solvedusing the dynamic inflow theory of Ref. 12. It as-sumes that the velocity normal to the rotor disk can

be expressed in terms of the radial and azimuthalexpansion functions,

w(s, ψ, t) =∞∑

r=0

∞∑

j=r+1,r+ψ,...

φrj(s)×

[αr

j(t)cos(rψ) + βrj (t)sin(rψ)

](7)

where s and t are the non-dimensionalized radiusand time, while ψ is the azimuth. The inflow states,α and β, are coefficients of terms containing theproduct of azimuthal harmonics and radial expan-sion functions φ, indexed by r and j, respectively.The governing equations for the inflow states aregiven by:

[M1]αrj+ [Lc]−1αr

j =12τmc

n [M1

] βrj + [Ls]−1βr

j =12τms

n (8)

The pressure coefficients, τ , are determined fromthe circulatory lift, which is available from eqn. 4by neglecting terms associated to accelerations inthe zeroth-order loading. The left hand side coef-ficient matrices, [M1] and [Lc,s] are determined bythe wake skew angle. Finally, with the solution ofthe inflow states, the zeroth-order inflow (λ0) is cal-culated from:

λ0 =∞∑

r=0

∞∑

j=r+1,r+3,...

J0

(rb

s

)φr

j(s)×[αr

j(t)cos(rψ) + βrj (t)sin(rψ)

](9)

where J0 is the Bessel function of the first kind andcan be approximated by taking the first few termsin its series.

Drag and Dynamic Stall

A potential benefit of camber actuation is theability to alter profile drag and stall characteristics,which have implications in power and vibration. Toinclude these effects in the low-order model, thepotential-flow airload expressions are augmentedwith a quasi-static profile drag term as well as adynamic-stall correction that is based on the ON-ERA model [13], taking the approach described inStumpf and Peters [14]. In the current implementa-

tion, the total airloads can be expressed as:

D0,tot = D0 + ρbcd(u20 + νT Sν)

12 u0

L0,tot = L0 − ρbcd(u20 + νT Sν)

12 Sν + ρu0Γ0

L1,tot = L1 + +ρu0Γ1 (10)

where cd is the profile drag coefficient, Γ0 and Γ1

are the dynamic stall states corresponding to the ze-roth and first-order generalized loads, respectively.The ONERA model assumes that the dynamic stallstates are governed by a second-order differentialequation, and requires static loading coefficientsnear and beyond stall. These, along with cd, are de-termined using the two-dimensional boundary layeranalysis code XFOIL [15] (which is valid to just-after-stall), along with a simple, empirically-derivedapproximation for deep-stall. The coefficients areobtained under varying Reynolds number, angle-of-attack and camber deformation. A detailed accountof the method used for determining the coefficientsis available in Thepvongs et. al [16].

Coupling with Finite-State Aerody-namics

The finite-state aerodynamics formulation usesChebyshev polynomials to form a basis for the cam-ber deformations and associated airloads, while thechoice of basis functions for the finite-section modesare arbitrary. The motion and force variables of theaerodynamics formulation are related to those of thestructural formulation by a simple linear expression,as derived in Thepvongs et. al [16]. This straight-forward connection between the aerodynamic andstructural states allows the same space and time in-tegration methods to be used for both formulationsas well as a simultaneous solution.

The governing structural dynamics equations(eqn. 1), aerodynamic load expressions (eqns. 4, 5),dynamic stall equations and wake equations (eqn. 8)together define the time-domain problem. An ex-plicit method is used with iterative refinement toachieve the desired convergence. Consider the fol-lowing form of the structural dynamics equations,obtained by applying spatial discretization (finite-elements) to eqn. 1. The set of differential-algebraicequations, along with the appropriate set of bound-ary conditions can be written as:

A(Xp)Xp + S(Xp, Xp) = LFS(Xp, Xp, Xp,

α |p, β |p, Γp)

BC(Xp) = 0 (11)

where A is the inertia matrix operator, S is thestructural matrix operator, written as functions ofthe structural state vector, X , its time derivative,X, and boundary values, X. These, along with theaerodynamic states, are all obtained at the currenttime (time is indexed by p).

The load matrix operator LFS is the contribu-tion of the aerodynamic loading which is a func-tion of the structural, inflow, and dynamic stallstates. A simple three-point backwards Euler time-integration scheme is used in accordance with thefirst-order form of the structural and potential flowgoverning equations (the ¨h term of eqn. 4 is obtainedby first derivatives of the velocity, which is a struc-tural state variable due to the mixed formulation).A four-point scheme is used to integrate the second-order dynamic stall equations. The time-integrationformulas are:

()p =3()p − 4()p−1 + 1()p−2

2∆t

()p =2()p − 5()p−1 + 4()p−2 − 1()p−3

(∆t)2(12)

The nonlinear system is solved, along with theboundary conditions of eqn. 11 by Newton-Raphsoniterations, with the Jacobians of the left-hand sideoperators of eqn. 11 available in closed form.

Trim

The enforcement of vehicle equilibrium adds morevariables and constraints to the aeroelastic problem.The present work assumes a wind-tunnel setup,where the variables are taken to be the collective,sine and cosine components of the cyclic pitch, andequilibrium is represented by specifying values forthe time-averaged thrust, pitch and roll moments.An auto-pilot method has been described by Peterset. al [17], which makes incremental changes to thecontrol settings at every timestep. The control set-tings are governed by:

τ θ + θ = J−1 a(G− g)− bg − dg

Jkl ≈ 1T

∫ T

0

∂gk

∂θ1dt (13)

where θ = [ θ0 θ1c θ1s ]T is the control settings,and G and g are the target and current values of thevariables representing equilibrium (i.e., the thrust,pitch and roll moments), respectively. For the testcases run in this study, coefficients b, d, and τ areset to zero, and a = 1/T , where T is the period.The ”trimmability matrix” J can be approximatedby numerically-computed Jacobian, determined bystepping the controls and examining the responseat an instant one revolution later. It should benoted that the trim is solved externally to the aero-dynamic and structural equations, using the timeintegration scheme of eqn. 12. Since the body loadsare computed from structural variables, namely theroot reaction loads, the trim is independent of theaerodynamics. Thus, the same trim algorithm andimplementation is used for the low-order and CFD-based analyses.

High-Fidelity AerodynamicsFormulation

Unstructured or mixed-element CFD methodshave advantages over structured-based CFD meth-ods that can be exploited for rotorcraft analysis. Inparticular, rapid grid generation for complex con-figurations is usually cited as a major reduction inthe preprocessing stage. Additional advantages ex-ist because unstructured methods can model rotorswith fewer overset grids, resulting in less computa-tional expenditure for grid motion and potentially areduction in numerical errors via fringe and orphancomputations. The most physically correct anal-ysis requires that all surfaces be accurately mod-eled via the time-accurate Navier-Stokes equations,requiring that two frames of motion be modeledduring a single simulation. While several optionsexist, one of the most applied approaches utilizesa combination of overset grids to generate smallergrids around each rotor blade, which then rotatethrough a background grid that may include a fuse-lage, ground planes, or other important configura-tions. This overset approach has been used to cor-rectly capture the general experimental trends onseveral configurations by Hariharan [18], Potsdam et

al [3] and Duque et. al [19], O’Brien and Smith [20]and Abras and Smith [21].

Because of the potential of the unstructuredmethodologies for advanced rotor configurations, aswell as their ability to rapidly model rotor vehiclecomponents, an unstructured methodology has beenchosen to provide the aerodynamic simulation forthis effort. The unstructured methodology selectedfor this effort is the FUN3D code developed at theNASA Langley Research Center [22–24]. FUN3Dcan resolve either the compressible or incompress-ible RANS equations on unstructured tetrahedral ormixed element meshes. The incompressible RANSequations are simulated via Chorin’s artificial com-pressibility method [25]. A first-order backward Eu-ler scheme with local time stepping has been appliedto steady-state applications, while a second-orderbackward differentiation formula (BDF) has beenutilized for time-accurate simulations. A point-implicit relaxation scheme resolves the resulting lin-earized system of equations. The RANS equationsare resolved via non-overlapping control volumessurrounding each cell vertex or node where the flowvariables are stored. Inviscid fluxes on the cell facesare solved via Roe’s scheme [26] that splits the fluxdifferences. The viscous fluxes are computed witha finite volume formulation to obtain an equivalentcentral-difference approximation.

The FUN3D methodology was originally de-veloped for fixed-wing applications by NASA re-searchers. Georgia Tech researchers have extendedit for rotary-wing applications of interest, as de-scribed by O’Brien and Smith [20] and Abras andSmith [21]. O’Brien [27] has shown that FUN3D’ssteady incompressible formulation is not only robustfor low-speed flight regimes, but both compressibleand incompressible results comparable to structuredmethods are obtained when the grids are locallycomparable on the surface and boundary layers. Inaddition, O’Brien developed an overset formulationof FUN3D to handle mixed inertial-moving framesnecessary to model rotating wings. This formulationmakes use of the DiRTLib [28] and SUGGAR [29]libraries to compute the cell connectivity and pro-vide the hole cutting necessary with moving frames.Abras [30] has extended the FUN3D code furtherto provide full rotor articulation, as well as loose-coupling with a computational structural dynamicscode, assuming that the rotor is modeled as a struc-tural beam. These extensions form the basis for thethree-dimensional aeroelastic rotating wing simula-tions, extended further to include camber deforma-tions.

Modifications for Aeroelastic Cou-pling

While many of the components necessary to per-form rotating, overset computations were extendedby prior Georgia Tech researchers, there remaineda number of modifications that were necessary toaccomplish full surface deflections within a tightly-coupled framework. FUN3D is capable of handlingmultiple grids to define the rotor blade and, as amassively parallel simulation methodology, will de-compose each grid further as it parses the computa-tional load among a number of processors. The cur-rent coupling scheme uses subdivisions of the bladesurface nodes, dividing them into smaller groups (aslater described). In order to couple FUN3D, thenodes of each of the blade surfaces in FUN3D mustbe identified and sorted into the groups. The bladelocal coordinates of the surface nodes in FUN3D arecollected from each processor. The CFD nodes thatdefine the edges of the groups may belong to morethan one group, so the nodes are flagged to ensurethat the same surface node is not passed to includedin the coupling more than once. Another array con-taining the coordinates of the surface face centers iscreated and sorted in the same way.

As is the case with most CFD solvers, FUN3Dresolves the primitive variables of the flowfield us-ing a nondimensional system. NLABS, as is typicalfor CSD solvers, uses dimensional quantities. Thus,the pressure coefficient on each cell face that lies onthe blade surface is computed within the CFD code,and are then dimensionalized using the freestreampressure of the simulation. Similarly, spatial coordi-nates of each cell are dimensionalized and the areaof the CFD cell computed, so that a dimensionalforce is computed from the cell pressure and area.This force array is passed into NLABS, which findsthe structural deflections at each node location andpasses the new coordinates back to FUN3D, alongwith the orientation of the blade.

For the coupling, it is necessary to deform notonly the CFD surface, but also the CFD volumegrid. As illustrated in Refs. 21 and 30, large sur-face deflections from not only the elasticity of theblade but new trim or articulation commands cancause the blade near-body grid to encounter poorcell resolution. In order the minimize the volumegrid deformation, the elastic deflection of the bladeis included in the Euler angle rotations. For exam-ple, in this application, the pitch angle is definedas the control pitch angle plus 0.5 times the elastic

twist of the blade at the tip. Flap and lead/lag an-gles are defined using a straight line extending fromthe elastic axis at the root to the location of theelastic axis of the tip. Using the deformations andany control changes from NLABS, the pitch, flap,and lead/lag angles are extracted and updated inthe six degree-of -freedom motion equation. The re-maining elastic deformations are then applied to therotor surface. The near-field volume grid based onthe new surface node coordinates is then generated,and the SUGGAR library [29] is then accessed tocalculate new connectivity data. At this point theflow solver is allowed to advance to a new time stepat which point the process is repeated.

Time Coupling and Initializa-tion

Due to its computational cost, the CFD-basedanalysis uses a weaker coupling of the fluid andstructural solutions compared to that of the low-order model. Any coupling scheme that needs re-peated solutions of the CFD or modification of theCFD boundary conditions during a timestep wouldnot likely be affordable. Instead, the current ap-proach obtains separate solutions for the CFD andCSD analyses, where, within a given timestep, thecomputed state of one is held fixed during the so-lution of the other, using the most-recently avail-able information and no subiterations. It has beendemonstrated by prior work [31, 32] that for hoverand forward flight that a loose-coupling between theCFD and CSD codes is sufficient. Further, fixed-wing researchers, (e.g. Refs. 33, 34), have shownthat for the small time step size needed for a stable(uncoupled) CFD solution, that the resulting errorsare negligible.

To further reduce computational time, the low-order aerodynamics model supports the initializa-tion and trimming of the CFD-based aeroelasticanalysis. Long-time simulations are required forcomputation of the trim Jacobian and convergenceto a sufficiently periodic and trimmed response. Thelow-order model can be used to compute the Jaco-bian and provide a starting condition of the CFD-based analysis. The starting condition is obtainedby allowing the low-order model to run to con-vergence, then extracting the structural state andcontrol settings. These are used to initialize theCFD-based analysis, which is run with the low-ordermodel kept activated. The applied aerodynamic

Figure 1: Simulation flow.

loads are set to gradually transition from those ofthe low-order model to those of the CFD in orderto prevent disturbances that may be slow to decay.

The time-coupling scheme described above is for-malized by augmenting eqn. 11 so that the schemebecomes:

A(Xp)Xp + S(Xp, Xp) =

ζLCFD(Xp−1, Xp−1, Xp−1)+(1− ζ)LFS (14)

where LCFD represents the numerically-computed(non-linear) CFD loads (transformed and integratedto structural forcing variables, as described in thefollowing section) and ζ is the weighting factor thatcontrols the transition between the CFD and low-order loads (0 ≤ ζ ≤ 1).

A diagram showing the general operation of thecombined aeroelastic simulation is shown in Fig. 1.

Transformation of Fluid –Structure Variables

The fluid and structure wetted surfaces are dis-cretized separately according to the resolution re-quirements of the individual analyses. Thereforea transformation is needed to properly connectthe motion and force variables between the two.This transformation has to consider the presenceof camber-wise flexibility in addition to the usualbeam degrees-of-freedom. Although the finite sec-tion modes are treated as one-dimensional beamvariables, they are defined in a discrete way within

the cross-section analysis. This is accomplished byspecifying the modes’ contribution to the displace-ment at a selected number of grid points withinthe cross section (which are a subset of the to-tal grid points used to discretize the cross-sectionproblem). Additionally, the motion of these cross-section grid points is determined partly by warpingarising from sources other than the finite-sectionmodes. For these reasons, quantities at the CFDgrid cannot be directly expressed by (or integratedto) one-dimensional beam variables, as is often donewhen coupling CFD to conventional beam analy-ses [31,32].

An intermediate step involving the three-dimensional structural grid is required. From thisthree-dimensional structural grid, the displacementsare interpolated to the CFD grid finite-elementshape functions, as originally described by Pida-parti [35] and extended by [37]. A triangulationof the structural grid points forms a new set of ele-ments which are used solely for the interface. Next,for each structural interface element, all enclosedCFD nodes are identified. The criterion for en-closure is that there exists a vector normal to theelement that intersects both the element and thegrid point. Among the structural interface elementsmeeting this criterion, the one having the shortestdistance to the CFD node is used for interpolation.The displacement of this CFD node, uCFD

j , is de-fined by:

uCFDj = Nju

CSDi + Rdj (15)

where uCSDi is the column matrix of the displace-

ments at the nodes of the structural interface ele-ment i, and Nj is the row matrix containing the el-ement shape functions evaluated at the CFD node.The shape functions are linear for the 3-node el-ement. The last term in the above equation ac-counts for separation between the CFD node andthe element plane. It preserves the separation dis-tance and CFD node’s local coordinate within theelement throughout the deformation. The vectorfrom the element plane to the CFD node is dj , andthe matrix rotating the element normal from its un-deformed to deformed orientation is R.

The transformation of the forces from the CFDcell centers to the structural interface mesh uses thesame finite-element shape functions. The force at aCFD cell, FCFD

k , is distributed to the nodes of theenclosing structural interface element by the rela-tion:

FCSDi = NT

k FCFDk (16)

where Nk represents shape functions evalutatedat the cell center position k. Some CFD nodes maynot be enclosed by a structural interface elementdue to the boundaries of the CFD and CSD gridsbeing slightly offset. The present study attemptsto model the same geometry (the wetted surface) ineach analysis, therefore the offset is caused only bythe different discretizations in each grid or numer-ical precision errors. For the displacement of thesepoints, extrapolation is performed using a functionfitted to the structural interface grid:

uCFDj =

P∑

i=1

[γji(xCSD

i − xCFDj )2 + r2

] 12 (17)

where xCSDi and xCFD

j are the position of the struc-tural and fluid nodes, respectively. P is the totalnumber of structural nodes over which the extrap-olation function is formed, r is a shape parameter(Ref. 36) and γji are the computed weighting coef-ficients. Following the extrapolation, the displace-ments of these points are averaged with its neigh-bors to improve smoothness between interpolatedand extrapolated regions. Forces originating fromCFD cell centers not enclosed by any element aresimply lumped along with its moment contributionto the nearest structural node.

In the current study, the CSD and CFD meshesare subdivided into four domains, corresponding tothe tip, root, top and bottom surfaces of the blade.The interpolations (specifically, the searching) andextrapolations are done separately for each domainto reduce overall computational costs. The interpo-lation and extrapolation algorithms described aboveare implemented in modified versions of the work byauthors of Refs. 36 and 37.

It should be mentioned that the fluid and struc-tural problems should ideally form a closed system,as discussed by Smith [38]. The right-most term ofeqn. 15, which has been added as part of the currentwork, removes the conservation of work across theinterface that normally occurs with finite-element-based interpolation. This term, however, improvedaccuracy of the interpolations when the separationdistance between the CFD nodes and its enclosingelement became large, as discussed in the next sec-tion. Similarly, the method of extrapolation andintegrating forces for unenclosed CFD nodes does

not enforce conservation of work. Errors resultingfrom the addition/loss of energy through the trans-formations are presumed to be small. This shouldbe further investigated, however, in future studies.

Numerical Investigation: RotorModel and Discretization

Results presented in this study are based on ascaled BO105 rotor model. The rotor is hingeless,4-bladed, and has a 2-m tip radius. From 0.44Rto the tip, the blade has a 0.121-m chord and anNACA 23012 airfoil, modified with a trailing-edgetab. The rotation speed is 109 rad/s.

The CFD blade surface grid (Fig. 2) was con-structed from the detailed geometry provided inRef. 39. The moving-volume grid forms a rectangu-lar prism that encloses the region between 1-chordlength inboard of the root-most blade edge to 2-chord lengths beyond the tip, 1-chord length aheadof the leading edge to 1-chord length behind thetrailing edge, and 2-chord lengths below the bladeto 2-chord lenghts above it. A total of 896,241 nodesare contained in the moving-volume grid of eachblade. The background grid is a cylinder extending40 m above and below the rotor, with a diameter of80 m, and is comprised of 1,046,807 nodes.

The CSD discretization is defined by the distri-bution of finite-element nodes along the beam ref-erence line, along with grid points within the cross-sections at those nodes. For the reference line, thereare 43 elements, concentrated near the transition re-gion. Each cross section contains 80 grid points, dis-tributed along the wetted surface and concentratednear the leading and trailing edges. These gridpoints and their triangulation for the fluid-structureinterface is shown in Fig. 3. A summary of the fluidand structural discretizations is given in Table 1.

For the cross-section properties, integrated massand stiffness data for the conventional beamdegrees-of-freedom were available, however, detailedlayup information could not be obtained. The finite-section mass and stiffness were set to values [16]that prevented significant camber deformation dueto aerodynamic loads or dynamic response. Thiscreated a realistic condition for the initial valida-tion of the simulations.

(a) CFD surface grid

(b) Near-field blade grid

(c) Full CFD volume grid

Figure 2: CFD grids for the BO105 rotor.

Figure 3: CSD mesh for the BO105 rotor.

Table 1: Blade discretization summary (per blade).

CFD surface nodes 20398CFD surface cells 40972Structural interface nodes 3680Structural interface cells 6864Structural 1-D elements 43

Numerical Investigation: Re-sults

Verification of the Fluid – StructureTransformation

The basic methodology of the interface algorithmshas already been validated for a set of canonicalcases, as presented by Lee [37]. Several aspects ofthe interface are re-examined, using the grids forthe BO105 model currently being studied. To verifythe transformation of forces, a load was prescribedat the CFD cell centers that varied substantially inamplitude and orientation. Fig. 4 shows the dis-tribution of the amplitude along the top and rootsurfaces. These forces were then transferred to thestructural nodes through the interface, and inte-grated to find the total forces and moments actingon the blade (the latter about the hub center). Un-der this loading, the error in the global forces alongthe three axes was found to be less than 1×10−12%and for the moments, less than 0.01%.

A visual approach was employed to verify thedisplacement transformations. A sample displace-ment field was set up by applying transverse andfinite-section (camber-wise bending) forces alongthe structure, producing large displacements. Fig. 5shows the co-movement of the fluid and structuralwetted-surface nodes.

The necessity of the rotational term in the in-

Figure 4: Test load magnitude (N) at discretepoints on CFD grid.

Figure 5: Sample undeformed and deformed gridsincluding camber-wise bending response – smallpoints: CFD; large points: structural interface.

terpolation scheme of Eqn. 15 is demonstrated inFig. 6. Because the CSD grid is substantially coarserthan that of the CFD, the CFD-node-to-interface-element distance can become large when there isdiscontinuous geometry in the CFD grid. An ex-ample is the transition region of the blade, nearthe trailing-edge tab. Fig. 6 shows this region,where the view is from the root towards the tip,as indicated by the arrow in Fig. 2(a). A 20

rigid-body pitch rotation is applied to the struc-ture, and the interface transforms the CFD meshshown in Fig. 6(a). The rotational term in the inter-polation equations preserves the shape of the mesh(Fig. 6(b)), while neglecting the term causes the sur-face to become wrinkled (Fig. 6(c)).

Forward Flight

With verification of the fluid-structural interface,several characteristics of the aeroelastic response areexamined for a forward-flight condition. The ad-vance ratio considered is 0.25, and the shaft angleis −5. Following the wind-tunnel experiments ofRef. 39, the trim condition is 3100N thrust (Fz),zero pitching moment (My) and zero rolling moment

(a) Original

(b) With rotational term

(c) Without rotational term

Figure 6: CFD grid near trailing edge of transitionregion; a) original; 20 deg. pitch applied throughfluid-structure interface with (b) and without (c)rotational term in Eqn. 15.

(Mx), evaluated as time-averaged quantities. In thefirst set of results, the effect of the transition lengthbetween the low-order and CFD loads is examined.Three time functions are setup for the variation of(ζ) in Eq. 14. The first is a step function, whilethe second two increase linearly from 0 to 1, over ahalf and full revolution. In the following figures, thefirst revolution contains the response computed bythe low-order analysis after its convergence to pe-riodicity and trim. After this revolution, the CFDsolution begins along with the transition from thelow-order to CFD loads, using one of the three tran-sition methods.

Figure 7 shows the time-history of the instanta-neous, fixed-frame hub loads, while Fig. 8 shows thecontrol settings. Comparing the converged controlsettings of the low-order and CFD-based analyses(Table 2), there is a 0.6 difference in the collective,while the cosine and sine components differ by 0.2

and 0.3, respectively. The step transition from low-order to CFD loads produces the largest deviationfrom the previously-established equilibrium. Thisdeviation is reduced by the half-revolution ramp,then further by the full-revolution ramp. Despitethis perturbation, the convergence of both the con-trol settings and hub loads appear to take less timeoverall using the step transition. In examining thecontrol settings, the collective changes at a fasterrate with the step transition while also appearingto have a head start compared to the other tran-sition methods. The sine and cosine componentsof the pitch control also show earlier arrival at theconverged value, although the rate appears to beunaffected by the transition method. These re-sults suggests that allowing the trim algorithm toact earlier on the CFD-based aeroelastic response ismore important than avoiding disturbances causedby sudden transitioning. In other words, the damp-ing (which is dominated by aerodynamic loading) issufficient to decay the disturbances such that theyare insignificant well before trim is reached.

Table 2: Control settings in degrees at convergenceof trim and aeroelastic response.

CFD Low-orderθ0 10.8 10.2θ1c 0.7 0.857θ1s -2.5 -2.82

The second set of results compare the

10

10.5

11

θ 0 (de

g.)

0.0 0.5 1.0 revs.

0

0.5

1

θ 1c (

deg.

)

0 1 2 3 4 5 6 7−3.5

−3

−2.5

θ 1s (

deg.

)

Revolutions

Figure 7: Pitch control history, varying length oftransition between low-order and CFD loads.

2000

2500

3000

3500

Fz (

N)

0.0 0.5 1.0 revs.

−200

−100

0

100

Mx (

Nm

)

0 1 2 3 4 5 6 7

−200

−100

0

My (

Nm

)

Revolutions

Figure 8: Hub loads history, varying length of tran-sition between low-order and CFD loads.

−100

0

100

Fx (

N)

CFD Low−order

−200

0

200

Fy (

N)

0 0.25 0.5 0.75 13000

3100

3200

Fz (

N)

Revolutions

Figure 9: Hub forces comparison of converged low-order and CFD predictions.

periodically-converged, trimmed responses ofthe low-order and CFD analyses as an attempt toco-validate them. The fixed-frame hub forces areshown in Fig. 9, while the moments are shown inFig. 10. The rearward and starboard-side loads,Fx and Fy, show good agreement in frequency andphase of the response, though significant differencesare seen in the amplitudes, as well as the meanvalue of Fy relative to its amplitude. As expected,the mean values of Fz,Mx, and My approach thevalues enforced by the trim algorithm, however,large differences can be seen in most other aspectsof the time response. Also, in the torque (Mz), thelow-order prediction of the magnitude of the meanvalue is 13% smaller than that of the CFD whichif uncoupled from the other errors, may indicatethat improvements in the drag model are needed.Lastly, it is important to note that for all of thefixed-frame loads, the CFD generally predicts alarger vibratory amplitude.

The tip deflections are presented in Fig. 11. Thechordwise (positive in the leading direction), flap-wise (positive up) and pitchwise (positive nose up)responses are shown. Several similarities can beseen in the predictions from the two analyses. Inthe chordwise response, both results show a dom-inant 1/rev component and a similar phase. TheCFD loads produce a larger mean deflection in thelag direction, which agrees with the larger torquemagnitude found in the hub loads. The flapwise re-sponse also shows agreement in several aspects. Thepositive peak occurs near 270 azimuth (0.75 of the

−100

0

100

Mx (

Nm

)

CFD Low−order

−50

0

50

My (

Nm

)

0 0.25 0.5 0.75 1−400

−350

−300

Mz (

Nm

)

Revolutions

Figure 10: Hub moments comparison of low-order(converged) and CFD (latest rotation available) pre-dictions.

revolution), while a flat response is seen near 0 az-imuth. The CFD loads, however, appear to inducea more significant 3/rev component than those ofthe low-order model. The mean values from bothpredictions are similar. In the pitchwise response,the absolute rotation is shown (includes both elasticand rigid-body contributions). The CFD predicts amore negative overall rotation at the tip, despitethe increased collective pitch. This suggests thata larger nose-down aerodynamic pitching momentis computed by the CFD analysis, and this may bethe source of the requirement for increased collectivepitch.

Despite the differences in the aeroelastic responsepredicted by the two aerodynamics models, corre-lation in several aspects including control settings,tip flapwise response and some hub loads, appearto verify a correct implementation, especially giventhe vastly different approaches used in coupling theanalyses.

Computational Requirements

The coupled code is executed on a cluster of IBMPower5+ 1.9 GHz processors. For the high-fidelityanalysis, a total of 64 processors have been used:63 for the parallel FUN3D computations and onefor the serial computations which includes NLABSand SUGGAR. Execution takes about 4.5 minutesper time step (about 27 hours per revolution). Itshould be noted that a new version of SUGGAR,

−0.02

−0.015

chor

dwis

e (m

)

0.04

0.06

0.08

flapw

ise

(m)

0 0.25 0.5 0.75 1−2

0246

pitc

h (d

eg.)

Revolutions

CFD Low−order

Figure 11: Tip deflections comparison of low-order(converged) and CFD (latest rotation available) pre-dictions.

which has been parallelized, is pending and shouldimprove the execution speed. The combined low-order aerodynamics, structures and fluid-structuretransformations take several seconds of serial com-putations for each time step.

Concluding Remarks

A simulation for high-fidelity analysis of rotorswith actuated, flexible airfoils has been presented.It includes a quasi-three-dimensional CSD analysis,RANS CFD (FUN3D) analysis, as well as coupling,trim and initialization algorithms. Results were pre-sented for a scaled BO105 rotor that explored sev-eral aspects of the coupling in addition to the gen-eral time response. When using a low-order aero-dynamics model for initializing the structural stateand control settings, a sudden transition betweenthe low-order and CFD-based loads was found toproduce faster convergence to a periodic, trimmedsolution compared to that produced by a gradualtransition. A comparison of the aeroelastic responsepredicted by the low-order aerodynamics model andCFD showed agreement in trimmed control settings,as well as aspects of the tip deflections and fixed-frame hub-loads. Although further validations areneeded, these results serve as a basic verificationof the methodology for coupling FUN3D with thethree-dimensional CSD analysis.

Acknowledgments

This work is sponsored by the NationalRotorcraft Technology Center (NTRC) VerticalLift/Rotorcraft Center of Excellence (VLRCOE) atthe Georgia Institute of Technology and Universityof Michigan. Dr. Mike Rutkowski is the techni-cal monitor of this center. Opinions, interpreta-tions, conclusions, and recommendations are thoseof the authors and are not necessarily endorsed bythe United States Government. Computational sup-port for the NTRC was provided through the DoDHigh Performance Computing Centers at ERDCand NAVO through HPC grants from the US Army(S/AAA Dr. Roger Strawn).

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