[american institute of aeronautics and astronautics 49th aiaa aerospace sciences meeting including...

Investigation of Viscous Effects on the AeroelasticStability of Transonic Airfoils

Hugo Stefanio de Almeida ∗

Instituto Tecnológico de Aeronáutica, São José dos Campos 12228-900, Brazil

João Luiz F. Azevedo †

Instituto de Aeronáutica e Espaço, São José dos Campos 12228-903, Brazil

The present work has the objective of comparing the effects of using viscous and non-viscous CFD models for the determination of the aerodynamic operator in aeroelasticstability analysis. The problems of interest in the present case consider both conventionaland supercritical airfoil configurations for steady, unsteady and aeroelastic computations.The 2-D Euler and Navier-Stokes equations are the most complete models for the flowsof interest in this case. For Navier-Stokes simulations, turbulence closure is achieving us-ing the one-equation Spalart-Allmaras turbulence model. A centered spatial discretizationscheme with explicit artificial dissipation is used in a finite volume method in the presentCFD code. The mesh movement algorithm used in the present work moves the entire com-putational domain as a rigid body. Most of the analysis in the present paper is performedover aerodynamic hysteresis curves. Finally, the dynamic stability aeroelastic analysis isperformed for a typical section model based on the NACA 0012 airfoil at transonic flightconditions. This analysis is performed in the frequency domain, rewriting the equations ofmotion using state space and pulse transfer function methodologies.

Nomenclature

a Speed of soundC Convective operatorCl Total lift coefficient integrated over airfoil surfaceCm Total moment coefficient integrated over airfoil surfaceCFL Courant-Friedrichs-Lewy numberD Artificial dissipation operatorei Total energy per unit of volumeE,F Flux vectors in the (x,y) Cartesian directions, respectivelyEe, Fe Euler flux vectors in the (x,y) Cartesian directions, respectivelyEv, Fv Viscous flux vectors in the (x,y) Cartesian directions, respectivelyh Plunge degree of Freedomk Number of facesM∞ Freestream Mach numbernd Node index of face kp PressurePr Prandtl numberPrt Turbulent Prandtl numberq Velocity magnitudeQ Vector of conserved properties

∗Graduate Student, Department of Aeronautical & Mechanical Engineering, Departamento de Ciência e Tecnologia Aeroes-pacial, DCTA/ITA/EAM-E, Brazil. E-mail: [email protected].

†Senior Research Engineer, Aerodynamics Division, Instituto de Aeronáutica e Espaço, Departamento de Ciência e TecnologiaAeroespacial, DCTA/IAE/ALA, Brazil. Associate Fellow AIAA. E-mail: [email protected].

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American Institute of Aeronautics and Astronautics

49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition4 - 7 January 2011, Orlando, Florida

AIAA 2011-994

Copyright © 2011 by H.S. de Almeida and J.L.F. Azevedo. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

Re Reynolds numberRHS Residue or right hand side operatorS Surface of the control volumesi Number of faces of the i− th control volumeΩ Magnitude of deformation tensort Timeu, v Velocity components in the (x,y) Cartesian directions, respectivelyxEA Elastic axis position normalized by airfoil chordxt, yt Contravariant velocity components in the (x,y) Cartesian directions, respectivelyV Viscous operator∀ Volume of control volumeα Pitch degree of freedomΔL Mesh characteristic sizeγ Ratio of specific heatsρ Densityν, νt Dynamic and turbulent viscosity coefficients, respectivelyν Modified viscosityψ Generic variableτ Stress componentsSubscripti i-th volumek k-th edge shared by i-th and nb-th volumesnb nb-th neighbor of the i-th control volumeSuperscriptn current time stepn− 1 old time stepn+ 1 new time step

I. Introduction

Over the past few years, the Computational Aerodynamics Laboratory of Instituto de Aeronáutica eEspaço (IAE) has been developing CFD solvers for two dimensional and three dimensional steady viscousflows.1,3 Since the last decade, the research group has been working with unsteady and non-viscous flowsfor transonic aerodynamic problems.5 Afterwards, Ref. 7 simulated the same transonic cases, but with morerefined meshes. An anlysis of the mesh dependency for steady and unsteady results was the main objectiveof that work. Better results are obtained for steady and unsteady cases, provided that a sufficiently refinedmesh is used and, still, one observes a large dependency on mesh refinament in the final results, speciallyfor aerodynamic coefficient computations. Marques8 revisited the work of Oliveira6 with more detailedinformation and extended the analysis for flat plate cases. The present work intends to complement Marques’work,18 comparing his inviscid results with those from viscous turbulent solvers and extend the applicationsto supercritical profiles in which mesh and viscous dependency, even for steady results, is expected to belarger than for conventional profiles.

It is known that inviscid solvers can provide good results to steady flows over conventional airfoils flyingat transonic Mach numbers, as indicated in Refs. 9–11. The unsteady results obtained by the same authorscompare well with experimental data for some aerodynamic coefficients. However, the simulations in Refs.9–11 used conventional airfoils. Moreover, although lift coefficient results were in good agreement with theavailable data, the moment coefficient, Cm, presented poor results, as shown, for instance, in Marques8 andBatina et al.14,15 Raveh13 and Darracq et al.30 obtained good results for unsteady transonic cases with largeshock wave oscillations for some airfoils. They compared results obtained with diferent turbulence modelsand results agree very well with experimental data. It is noteworthy that even the moment coefficient, whichis more sensitive to viscous structures than the force coefficients, is in good agreement with the comparisondata. The motivation of the present work concerns the determination of aerodynamic coefficients using aviscous formulation for evaluation of the effect of viscous terms in the determination of aeroelastic stabilityin transonic flow conditions over supercritical airfoils.

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At transonic speeds, the well-known phenomenon called transonic dip occurs. It reduces, in a non-linearform, the velocity in which flutter occurs. It is strictly dependent on shock wave motion and the aerodynamiclags typical of the transonic regime. For supercritical airfoils, in which the transonic shock wave is delayed,it is expected that shock–boundary layer interaction can also play a major role in capturing the actualphenomenom. Hence, it is expected that the nonlinear and non-viscous formulation based on the Eulerequations cannot precisely establish the transonic dip phenomenon for such class of supercritical airfoils.The present work starts by demonstrating that the inviscid portion of the formulation is working properly.For that, simulations of both steady and unsteady flows are performed. In the latter case, the authorslook at hysteresis curves for the aerodynamic coefficients as a form of ascertaining that the results of Ref. 8can be reproduced. Afterwards, the same simulations will be repeated using a viscous formulation in orderto quantify the effect of the viscous terms in the unsteady aerodynamic results. It is expected that, forconventional airfoils, such viscous effects will be small for both lift and moment coefficients. On the otherhand, for supercritical airfoils, it is anticipated that such effects may play a major role in the final unsteadycoefficients.

The 2-D unsteady Reynolds-averaged Navier-Stokes (URANS) equations are the formulation implementedin the CFD solver developed by the research group. This code is written in a cell centered, finite volumecontext for unstructured meshes. The solver is able to handle any type of mesh, since it is built using aface–based (actually, edge–based in 2-D) data structure, in which properties are computed at faces and thecontribution to shared volumes is distributed accordingly. A hybrid, explicit, Runge-Kutta time marchingscheme is used to advance the solution in time. This scheme is written to support steady and unsteadysimulations. In the latter case, one must also provide support for the contribution of mesh deformation.Moreover, an implicit Euler method is used to solve the turbulence modeling equations, as previous expe-rience3 indicated that an implicit time integration is required to maintain numerical stability of turbulencemodels. Furthermore, the convective terms of the Spalart-Allmaras (SA)29 turbulence model are discretizedas a 1st-order upwind flux scheme to avoid adding explicit artificial dissipation contribution to maintain sta-bility of the solution. All other terms of the governing equations are spatially discretized using the equivalentof a central spatial discretization scheme plus added artificial dissipation.26,27

II. Theoretical Formulation

The problems of interest in the present work can be adequately represented by the 2-D Euler and/orNavier-Stokes equations. The 2-D URANS equations can be written in dimensionless and conservative formas

∂Q

∂t+

∂E

∂x+

∂F

∂y= 0 . (1)

E and F fluxes contain contribution of inviscid and viscous terms. These flux vectors can be written as

E = Ee − Ev ,F = Fe − Fv . (2)

The vector of conserved variables and the convective flux vectors are

Q =

⎧⎪⎪⎪⎨⎪⎪⎪⎩ρ

ρu

ρv

ei

⎫⎪⎪⎪⎬⎪⎪⎪⎭ , Ee =

⎧⎪⎪⎪⎨⎪⎪⎪⎩ρU

ρuU + p

ρvU

(e+ p)U + xtp

⎫⎪⎪⎪⎬⎪⎪⎪⎭ , Fe =

⎧⎪⎪⎪⎨⎪⎪⎪⎩ρV

ρuV

ρvV + p

(e+ p)V + ytp

⎫⎪⎪⎪⎬⎪⎪⎪⎭ . (3)

The contravariant velocity components, U and V , carry, for unsteady cases, the contributions from meshvelocities. The present approach initially computes the node mesh velocity components. However, sincethe code uses a cell centered method, such node velocity components are averaged in order to yield volumecentroid velocity components, xt and yt. Hence, U and V can be written as

U = u− xt , V = v − yt . (4)

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The viscous flux vectors are given by

Ev =M∞Re

⎧⎪⎪⎪⎨⎪⎪⎪⎩0

τxxτxy

τxxU + τxyV − qx

⎫⎪⎪⎪⎬⎪⎪⎪⎭ , Fv =M∞Re

⎧⎪⎪⎪⎨⎪⎪⎪⎩0

τxyτyy

τxyU + τyyV − qy

⎫⎪⎪⎪⎬⎪⎪⎪⎭ . (5)

The Reynolds number appears in the previous expressions from the equation nondimensionalization andrepresents a dimensional scale of the flow. In the present work, the Reynolds number is computed inreference to the airfoil chord, c, and the freestream velocity magnitude as

Re =ρ∞q∞c

μ∞. (6)

The q∞ variable is the velocity magnitude in the freestream. For a 2-D code, it is defined as√u2∞ + v2∞.

The problem requires some constitutive relations in order to have equal number of equations and the numberof flowfield variables. Hence, pressure is computed by the equation of state for perfect gases as

p = (γ − 1)

[ei − 1

2ρ(u2 + v2)

]. (7)

In the previous equation, γ is the ratio of specific heats, and it taken as 1.40 for all calculations hereperformed, as usual for air. The viscous stress tensor and the heat flux vector, written in Einsten’s indicialform, are given by

τij = (μl + μt)

[(∂ui

∂xj+

∂uj

∂xi)− 2

3

∂un

∂xnδij)

], qj = −γ( μl

Pr+

μt

Prt)(∂ei∂xj

) . (8)

Usually in URANS codes, the contribution from small turbulence scales of time, space, and vorticity areadded explicity into the flux vector and stress tensor. As can be seen in Eq. 8, this contributiton is formedby adding the so–called turbulent viscosity coefficient, μt. This variable assumes a zero value in frestreamregions, once no turbulence is generated, to higher values near specific regions of the domain, usually nearwalls, in boundary layers, and in the airfoil wake in aerodynamic problems.

III. Spatial Discretization

The CFD solver used in the present work was largely used and validated in inviscid8 and viscous1simulations. This solver was build in a centered finite volume context. The finite volume method performsan integration of Eq. 1 in each control volume of the flowfield. After integration and application of Green’stheorem, one obtains

∂

∂t

∫∀Qd∀+

∫S

(Edy − Fdx) = 0 . (9)

The discrete vector of conserved variables for the i− th control volume, Qi, is defined as

Qi =1

∀i

∫∀i

Qd∀ . (10)

Now, the problem is how to obtain the flux through the faces of each control volume of the discretedomain. Following the work of Mavriplis,23 flux computations at the face are given by∫

si

(Edy − Fdx) ≈ C (Qi) =

si∑k=1

[E (Qk) (y2 − y1)− F (Qk) (x2 − x1)] . (11)

Flux contributions E (Qk) and F (Qk) are computed with the prescribed value in the face, Qk, so thiscontribution is added to the i − th volume and substracted from its neighbor, nb − th control volume.Furthermore, any property, which is needed at the face, must be estimated as a simple average between the

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adjacent volumes, i and nb, which share the k − th face. For a general variable at the k − th face, one canwrite

ψk =ψi + ψnb

2. (12)

The viscous flux derivatives are also compute by a surface integration over the control volume faces, accordingto Ref. 22.

Some artificial dissipation is required in order to maintain the numerical stability of the simulation.Once a centered spatial discretization scheme is used, explicit addition of artificial dissipation is requiredbecause a centered scheme is, by construction, undamped and, therefore, it allows the generation of numericalinstabilities. Those instabilities appear, mainly, as oscilations in regions of strong gradients, such as in shockwave regions. Mavriplis23 has proposed the following artificial dissipation scheme

D (Qi) = ξ2i,nbd2 (Qi)− ξ4i,nbd

4 (Qi) . (13)

The d2 and d4 operators are, respectively, the undivided Laplacian and biharmonic operators. The firstterm is related to damping numerical instabilities in the presence of shock waves whereas the second term isresponsible for the background numerical dissipation. These operators are defined as

d2 (Qi) =

si∑k=1

[(Ai +Anb

2

)(Qnb −Qi)

],

d4 (Qi) =

si∑k=1

[(Ai +Anb

2

)(∇2Qnb −∇2Qi

)]. (14)

The ∇2Qi terms of the biharmonic operator may be obtained as

∇2Qi =

si∑k=1

(Qnb −Qi) . (15)

The weighting terms for the artificial dissipation are based on the spectral radius of Euler equation in thedirection normal to the control volume edges. Hence, the Ai terms are built as

Ai =

si∑k=1

(|ukΔyk − vkΔxk|+ ak

√Δx2

k +Δx2k

). (16)

The biharmonic operator is 3th-order accurate in space, and the summation is performed over the edges ofthe control volume. The contribution of the ∇2Qi operator is neglected at boundaries and, also, near regionsof large flowfield gradients. In particular, near large gradient regions, this term is automatically zeroed outby the pressure sensor test. This sensor identifies the regions in the flowfield in which the pressure gradientis high. The pressure sensor, (νi), is defined as

νi =

∑sik=1 | (pnb − pi) |∑sik=1 (pnb + pi)

. (17)

For ghost volumes, the pressure sensor assumes the same value of the interior volume which shares the k− thface. The νi term is responsible to alternate the dissipative contribution between the d2 and d4 operators,according to the following rationale:

ξ2i,nb = K2 max (νi, νnb) , ξ4i,nb = max(0,K4 − ξ2i,nb

). (18)

The reference values suggested by Mavriplis23 for the K2 and K4 contants are

K2 =1

4, K4 =

3

256. (19)

The maximun eigenvalue of the Euler equations intends to consider only the worst case. However,considering the worst case means that, for less critical cases, less dissipation per iteraction would be necessary.Hence, artificial dissipation schemes could be built to take such situation into account. Those schemes areextensively discussed in Refs. 4, 21. Moreover, in Ref. 4, several artificial dissipation schemes, including thepresent one, are discussed and compared with each other for some transonic test cases.

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IV. Temporal Discretization

The temporal scheme used is a 2nd-order, 5-stage, explicit Runge-Kutta (RK5) time marching scheme,described, among other references, in Refs. 26,27. In order to take into account the unsteady flow phenomena,Batina16 added the contribution of the ratio of control volume deformation. This contribution is felt in termsof induced edge velocities and volume variation. This volume variation is explicitly taken into account in theRK5 scheme as a ratio between the values of the i− th control volume area in two consecutive time steps.

Q(0)i = Qn

i ,

Q(j)i =

∀ni∀n+1i

Q(0)i − αj

Δti

∀n+1i

[C(Q

(j−1)i

)− D

(Q

(j−1)′

i

)− V

(Q

(0)i

)], j = 1, . . . , 5 , (20)

Qn+1i =

∀ni∀n+1i

Q(5)i .

The residue operator, RHS, computed for the i − th control volume, is composed by the summation ofconvective terms, artificial dissipation operator and viscous terms. Hence, it can be written as

RHSi = −[C (Qi)− D (Qi)− V (Qi)

]. (21)

The artificial dissipation operator is computed only in the first two stages for inviscid simulations. There-fore, the (j−1)′ superscript in Eq. 20 is equal to 0 in the first time step and equal to 1 in the others. Otherwise,for viscous simulations, the artificial dissipation operator is computed at alternate stages. This approachleads to a low computational cost by stage and guarantees good numerical stability. The α coeficients usedin the present work are

α1 =1

4, α2 =

1

6, α3 =

3

8, α4 =

1

2, α5 = 1 . (22)

This scheme is known as a hybrid scheme because the convective, dissipative and viscous fluxes are notupdated in the same stages within a time step.

V. Convergence Acceleration Techniques

Unsteady simulations require a physically sound initial solution. This steady solution must be converged.Therefore, it is important to have convergence acceleration techniques which would expedite convergence tothe steady state solution, that is, the initial condition for the unsteady simulations. Convergence is achievednot only when the residue of the URANS equations approach machine zero, but also when the aerodynamiccoefficients become constant along the time integration. The convergence of the steady aerodynamic coef-ficients is considered by the authors, because a long simulation run is required to achieve machine zero forthe RHS operator. On the other hand, the force coefficients remain unchanged, or oscillanting with verylow amplitude around a constant value, while the residue tends to zero. A typical behavior for aerodynamiccoefficients and residue convergence analysis can be seen in the left and right plots of Fig. 1, respectively.Observe that, before the first ten thousands iterations, coefficients of moment and lift change very fast. Thisis the initial (numerical) transient and cannot be considered as a physical solution. From this point on, themagnitudes of the coefficients achieve the correct values as the iterations progress. Hence, the desired steadystate solution is achieved. This steady solution is necesssary as the initial condition for unsteady harmonicforced simulations.

Some techniques are used to accelerate convergence in the present work. It is important to emphasizethat the techniques here implemented can only be used if, and only if, a steady flow is expected. Hence, it isconceptually unacceptable to use these techniques in unsteady flows. On the other hand, if a dual-time-stepalgorithm were used in the current simulation, all inner iteraction can take advantage of these techniquesto accelerate convergence. The approach concerns the use of a constant CFL number as can be seen in Ref.24. Hence, different local time steps may be obtained for different control volumes. The local time step isproportional to the volume of the i − th control volume. Furthermore, it considers the maximum possibleeigenvalue for the Jacobian matrices for the Euler equations, λi = |q|+a, where |q| is the velocity magnitude,as a form of estimating the propagation speed. An expression for the variable time step is built as

Δt =CFL ΔLi

λi. (23)

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Iteractions

Coe

ffici

entV

alue

0 50000 100000

-0.1

0

0.1

0.2

0.3

ClCm

Iteractions

Log(

RH

S)

0 50000 100000

-10

-5

0

Figure 1. Convergence of steady solution based on aerodynamic coefficients and maximun residue analysis,respectively.

The ΔLi is defined in the present work as the radius of the inscribed circle in the i − th control volumeand CFL is the "Courant-Friedrichs-Lewy" number, that represents how fast the information is propagatedthroughout the domain. The CFL number is maintained contant for all volumes of the domain during asteady state simulation.

Some other techniques were used in the present effort, such as an FAS multigrid algorithm23 and animplicit residual smoothing procedure.27 Those techniques, once coupled to the constant CFL approach,increase considerably the convergence rate. Moreover, the authors suggest a multigrid constant CFL cor-rection to be applied to coarser meshes in order to increase the stability margins of the simulations, furtherimproving the convergence rate of the simulation. The proposed equation is implemented as

CFLnm+1 =1

ξnm+1CFLnm , (24)

where nm is the index of the nm-th mesh in the multigrid scheme indexation. Moreover, ξ is the agglom-eration factor, which represents the area ratio between two adjacent multigrid levels. One can compute itas

ξnm+1 = min

(ξ0,

∀nm∀nm+1

). (25)

Finally, ξ0 is set by the user in order to perform a relaxation on the information exchange.

VI. Dual Time Step Technique

A dual time step algorithm was implemented in the aeroelastic solver. This technique was implementedbecause unsteady viscous simulations were requiring time steps which were extremely small, especially forturbulent flows. The dual time step technique was implemented in accordance to Jameson’s work.25 As itis well-known, the technique solves a steady simulation in pseudo-time for each real time step. Moreover,Hsu and Jameson10 proposed the use of a set of improvements in the original work, in order to increase theconvergence rate for unsteady simulations. The latter authors have used an implicit modified ADI algorithmin order to achieved second order accuracy in time. A flowchart, shown in Fig. 2, indicates how the dualtime step technique was implemented in the present solver. Implementation of this technique leads to thesolution of several steady simulations for each update in the aeroelastic DOFs. Furthermore, all convergenceacceleration techniques, previously discussed, can be used in order to expedite convergence of the dual-timeproblem.

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Figure 2. Flowchart of the dual time step algorithm.

VII. Mesh Deformation Algorithm

The methodology used in the present work considers the airfoil as a rigid body. However, the physicalairfoil is a continuous and flexible body with a infinite number of degrees of freedom (DOFs). The structuralmodel must represent, in a certain way, this natural airfoil flexibility. For the typical section model,17 theairfoil flexibility is represented by two DOFs, i.e., those for airfoil pitching and plunging. These DOFs tryto represent a rigid airfoil with the features of a flexible one.

The harmonic external excitation, which is performed as an input for the structural model, induces aharmonically oscilatory response of the airfoil aerodynamic coefficients. This type of harmonic movimentcan be seen as a forced motion imposed to the airfoil, in order to study its behavior in a certain reducedfrequency of interest. The present work has only considered oscillations in pitch, since the effects are deemedmore pronounced for the present applications and it is easier to find comparison data in the literature. Hence,the computational mesh over the airfoil must move accordingly.

In the present case, the option was for moving the mesh as a rigid body attached to the airfoil. Thisrigid body algorithm considers no relative motion between the airfoil nodes and the mesh nodes. Hence, thegrid points translate and/or rotate following the harmonic forced airfoil movement. Each grid point has thesame type of motion. The grid points move according to

xn+1i = (x0 − xEA) cos (α(t) − α0) + y0 sin (α(t) − α0) +

xEA

2,

yn+1i = y0 + cos (α(t) − α0) + (x0 − xEA) sin (α(t) − α0) + h (t) . (26)

For this algorithm, the outer boundary edges move as a rigid body together with all interior grid points.Therefore, there is no relative movement among the nodes. One should observe that the airfoil movementrelations always refer to the initial mesh. Such an approach reduces the possibility of cumulative numericalround-off errors for the mesh movement algorithm. These numerical errors are particularly important forturbulent meshes, because the grid is extremely refined near the bodies in order to resolve the turbulence

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modeling equations. In order to reduce such type of errors, the aeroelastic viscous code is written usingdouble precision for all cases.

VIII. Turbulence Model

Turbulence is a natural phenomena with a large spectrum of scales, as indicated in Ref. 28, that can bepredicted by the Navier–Stokes equations. Turbulence phenomena appear as a result of the Navier–Stokesequations, and they do not require an extra equation to solve them. However, solution of all turbulencescales needs large amounts of computational resources, since an extraordinarily refined mesh is necessary.The direct solution of the turbulent scales is known as DNS, for Direct Numerical Simulation. Hence, forthe large majority of engineering applications, DNS is beyond the range of the computational resources.

In order to predict the turbulence phenomena with some reasonably sized mesh, the present work usesthe URANS formulation. In this formulation there are nonlinear new terms that apppear from the averagingof the equations. Such new terms are modeled by turbulence models. For the methodology used here, incompressible aerodynamic solvers, these new terms of URANS modeling are rewritten as single functionsof the turbulent viscosity, νt, and the turbulent Prandtl number. These types of turbulent models areknown as linear eddy-viscosity models. The turbulence model by Spalart and Allmaras,29 SA, is used in thepresent simulations. This is a robust model which requires less refined meshes near wall regions, y+ = 1, incomparison to other models, which are typically two-equation models, such as Menter’s SST34 and the lowReynolds κ− ω35 models.

For the SA turbulence model the eddy viscosity νt is constructed as

νt = νfv1 , fv1 =X3

X3 + c3v1. (27)

The X variable is defined asX =

ν

ν. (28)

The differential equation for the evolution of ν is given by

Dν

Dt= cb1Sν +

1

σ[∇. ((ν + ν)∇ν) + cb2 (ν)]− cw1fw

(ν

d

)2

. (29)

The term of turbulence production, i.e., the first term in the right-hand side of Eq. (29), is built taking intoaccount the contribution of the gradients of the flowfield variables. The S term is defined as

S = Ω+ν

k2d2fv2 , fv2 = 1− X

1 +Xfv1. (30)

The d variable is the distance from the control volume centroid to the nearest wall, k is the von Kármánconstant and assumes the value of 0.41. Further, the S term represents the turbulence production due tovelocity gradients and it is also dependent on the distance to the wall, d. This production term is responsiblefor generating the turbulence near wall regions. In the wake region, the contribution of d is almost neglectedand the turbulence generation is dominated by the contribution of the velocity gradients. The destructionterm, the last term in the right-hand side of Eq. (29), is also proportional to the distance to the nearestwall. The destruction term is proportional to 1

d2 , and it avoids an unbounded turbulence generation. Thefv2 term also controls the turbulence production, weighting it in the [0, 1] range. In the free stream region,fv2 is zero. The second term in the brackets in the right-hand side of Eq. (29) is composed by the diffusiveand the cross-diffusion terms, respectively. The model constants are

cb1 = 0.1355 , cb2 = 0.622 , σ =2

3, cv1 = 7.1 , cw1 =

cb1k

+(1 + cb2)

σ, cw2 = 0.3 , cw3 = 2.0 . (31)

The fw function is built as

fw (r) = g (r)

[1 + c6w3

g6w3 + c6w3

] 16

. (32)

The other relations for g and r are

g (r) = r + cw2

(r6 − r

), r =

νt

Ωk2d2. (33)

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This version of the Spalart-Allmaras turbulence model neglects all terms which deal with the transitionphenomenon. Therefore, the flowfield is considered fully turbulent throughout the domain.

IX. Aeroelastic Analysis Methodology

A. General Formulation

The structural formulation, used in the present work, represents the airfoil motion in two degrees of freedom.This structural model is based on the classical typical section model.17 This model considers two DOFs forairfoil movement, namely pitch and plunge, i.e., α and h, respectively. These DOFs attempt to represent athree-dimensional wing by an effective 2-D section at 75% semi-span from the wing root. The typical sectionmathematical model can be obtained using Lagrange’s equations of motion17 as

mh (t) + Sαα (t) + khh (t) = Qh (t) ,Sαh (t) + Iαα (t) + kαα (t) = Qα (t) , (34)

where the static unbalancing and polar inertial moment are defined as

Sα = mbxα and Iα = mb2r2α . (35)

In the previous relations, m is the airfoil mass, rα is radius of gyration, and kh and kα are the heaving andtorsional spring coefficients, respectively. Rewriting the system in dimensionless form, one can obtain

η (t) + xαα (t) + ω2hη (t) =

Qh (t)

mb,

xαη (t) + r2αα (t) + ω2αr

2αα (t) =

Qα (t)

mb2. (36)

This approach is used to explicitly indicate the natural frequencies of the aeroelastic system. Moreover, theaeroelastic system is dependent on a few parameters. These parameters govern the system response andaeroelastic stability. A study of their influence is one of the main objectives of the present work. Hence,system parameter influence is added to the present work in order to perform a sensitivity analysis.

Rewriting the system of structural equations, Eq. 36, in matrix form, one can obtain that

[M ] {η (t)}+ [K] {η (t)} = {Q (t)} . (37)

Hence, the typical section theory can be written and easily implemented as[1 xα

xα r2α

]{η (t)}+

[ω2h 0

0 r2αω2α

]{η (t)} = {Q (t)} . (38)

The coupling of the aeroelastic system for both pitch and plunge DOFs is observed in the mass matrixformulation. The xα parameter is responsible for this behavior. This parameter is the distance from theairfoil mid-chord point to the airfoil elastic axis. According to this parameter, the aeroelastic system can bestable or not. The airfoil free vibration frequencies, for plunging and pitching DOFs are, respectively,

ω2h =

Kh

mand ω2

α =Kα

Iα. (39)

The vectors of forces and generalized coordinates behave as a linear combination of pitching and plungingaeroelastic DOFs as

{η} ={

η (t)

α (t)

}, {Q (t)} =

{Qh

mbQα

mb2

}. (40)

Using the hypothesis of linear behavior of the aerodynamic coefficients with respect to the airfoil DOFsmotion,17 one can write for lift and moment coefficients

L = Qdccl = Lhh+ Lαα

My = Qdc2cm = Mhh+Mαα . (41)

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Using a linear decomposition of the aerodynamic input, one can obtain that

cl =L

Qdc=

Lhh

Qdc+

Lαα

Qdc, (42)

cm =My

Qdc2=

Mhh

Qdc2+

Mαα

Qdc2, (43)

where Qd is the dynamic pressure, which is defined as

Qd =1

2ρU2

∞ . (44)

Moreover, the generalized forced vector components are written as

Qh = −L = −Q (2b)

(clh

ξ

2+ clαα

),

Qα = My = Q (2b)2

(cmh

ξ

2+ cmαα

). (45)

Finally, Eq. 37 can be rewritten as

[M ] η (t) + [K] (t) =U∞πb2ζ

[A] η (t) , (46)

where ζ is the mass ratio defined as:ζ =

m

πρb2. (47)

This parameter relates to the cylindrical mass ratio around an airfoil of chord equal to 2b.The aerodynamic influence coefficient matrix, which directly influencee the system response, is defined

as

[A] =

[−Clh

2 −Clα

Clh 2Cmα

]. (48)

Rewriting Eq. 46 in dimensionless form, one can obtain

[M ] η (t) + [K] (t) =U∞πb2ζ

[A] η (t) . (49)

In order to adjust the entire system parameters to the dimensionless form, the system time parameter shallbe rewritten as a function of the reference frequency, ωτ ,

t = ωτ t , (50)

This is the reference frequency which, in the present work, is considered equal to airfoil free vibrationfrequency in the pitch mode, i.e.,

ωτ = ωα . (51)

Hence, in dimensionless form, Eq. 49 is rewritten as a function of the dimensionless time as

[M ] η(t)+[K] (

t)=

(U∗)2

πζ[A] η

(t)

, (52)

where,[K]= 1

ω2r

[K]

is the new stiffness matrix in dimensionless form. The dimensionless velocity, U�, is

U� =U∞bωτ

=1

κr. (53)

The previous expression indicates that the reference reduce frequency of the system, κr, is computed as afunction of the reference frequency, ωτ .

The coefficients of the aerodynamic influence matrix can be calculated in the frequency domain from animpulsive motion, such as the exponentially-shaped pulse function.15,18 Hence, the number of aerodynamiccomputations, for each flight condition, reduces to an unsteady run for each structural mode, i.e., pitchingand plunging once using a typical section approach. The hypothesis of the linearity of aerodynamic coefficientbehavior with respect to the airfoil DOFs motion is valid for both viscous and inviscid flow fields.

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B. Transfer Function Analysis

The CFD solver is used to obtain the aerodynamic response from an impulsive input in each of the DOFs ofthe typical section model. Moreover, the outputs in terms of Cl and Cm coefficients are used to transformthe aerodynamic response, in the time domain, to a set of aerodynamic responses in the Laplace domain bymeans of applying a FFT procedure to the unsteady aerodynamic responses. Furthermore, the responses inthe Laplace domain are used to construct the matrix of aerodynamic coefficients in the frequency domain.18Finally, the aeroelastic eigenvalue problem is solved in the Laplace domain to obtain the final aeroelasticstability characteristics of the system.

The main structure of the aeroelastic solver can be viewed in six steps:

1. Select a test case of interest, i.e., specify a configuration and the desired flight condition (typically byspecifying Reynolds and Mach numbers);

2. Solve the steady aerodynamic problem for such configuration, using viscous or inviscid formulations;

3. From the steady result of the previous item, the unsteady problem is solved. This stage uses animpulsive method to excite the airfoil DOFs. The excitation is performed separately for each structuralmode;

4. Obtain the FFT of the temporal history of the aerodynamic coefficients. The definition of reducedfrequency axis, κ, is performed as

κ =ωc

2U∞=

πfc

U∞, (54)

f = gj

N, j = 0, 1, 2, ... , (N − 1) , (55)

where N is the total number of temporal points in the coefficient time history,

g =1

Δt=

a∞cΔt

, (56)

andκi =

π

M∞Δt

j

N, j = 0, 1, 2, ... , (N − 1) ; (57)

5. Approximate the tabulated values obtained in the previous item by using an interpolating polynomialfunction;18

6. Finally, the [B] matrix of the aeroelastic system is constructed. This matrix contains all informationfrom the unsteady flow field history and the structural parameters. An eigenvalue analysis of theaeroelastic matrix determines the aeroelastic system behavior. Moreover, the number of aerodynamiclag states and the range of reduced frequencies of interest are considered in this matrix.

X. Inviscid and Viscous Terms Aeroelastic Solver Results

In this section, the investigation of the behavior of the aeroelastic CFD solver is perfomed for some testcases. All simulations are evaluated at transonic Mach numbers, which is the most interesting range of flowspeeds in the present effort.

A. Validation of Viscous Terms of the Aeroelastic Solver

NACA 0012 and NACA 64A010 conventional airfoils are considered for some fligth conditions. The main flowphenomena of interest in these cases are the transonic shock wave over the airfoil, the turbulent boundarylayer and the mutual interactions between both nonlinear phenomena. The test case of the NACA 0012airfoil is considered with zero angle of attack and M∞ = 0.8. The Reynolds number is 10 × 106, based onthe airfoil chord length. The experimental results of Ref. 39 are considered for comparison of the presentsimulations. Figure 3 shows the Euler mesh generated around the NACA 0012 airfoil and the main resultsconsidering an inviscid formulation for such test case. One can observe in Fig. 3 that the code is capturing

12 of 30


X

Y

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

X

Y

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

Mach

1.110.90.80.70.60.50.40.30.2

X

Y

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

P

1.0510.950.90.850.80.750.70.650.60.550.50.450.40.350.30.250.2

Figure 3. Euler mesh around NACA 0012 airfoil, and inviscid results in terms of Mach number and dimen-sionless pressure contours, respectively (M∞ = 0.8, Re = 10× 106, and zero angle of attack).

correctly the transonic shock wave over the airfoil.The shock wave characterize itself by a large discontinuity of flow field variables, such as Mach number,

density and pressure. However, using a viscous formulation, the intensity of the shock is decreased becausethe viscous influence of the boundary layer absorbs some energy from the flow field. Figure 4 compares,in terms of the pressure coefficient, the inviscid and viscous results. The mesh for viscous calculationswas developed such that 70 control volumes are created within the boundary layer. Results also suggestvery good agreement of shock wave and boundary layer computation using the viscous formulation. Thisgood agreement between experimental data and viscous results demonstrates the robustness of the SAturbulence model to predict the shock wave–boundary layer interaction. This phenomena is triggered bysome continuous feeding of energy from the flow field which initiates instabilities of vorticity turbulencescales, and their interaction extracts energy from the flow. This energy is used to increase the vorticity anddecrease the energy and velocity turbulent scale magnitude. In the present simulation, for the Mach andReynolds numbers considered, the boundary layer does not separate. The separation does not occur eitherat the shock front or in the leading edge region, as one can observe in Fig. 5. Therefore, there is a balancebetween energy and vorticity scale magnitude in order to avoid an unbounded enlargment of vorticity scales.In Fig. 5, one can further observe that there is no reverse flow region over the airfoil surface.

The flow over a NACA 64A010 airfoil at transonic conditions is also simulated in order to furtehr evaluatethe viscous solver. Figure 6 presents simulated results obtained for this airfoil in terms of aerodynamic

13 of 30


x/C

Y

0 0.2 0.4 0.6 0.8 1

-0.4

-0.2

0

0.2

0.4

0.6

Experimental Data

x/C

-Cp

0 0.2 0.4 0.6 0.8 1-1.5

-1

-0.5

0

0.5

1

Viscous Simulation

+

+

++

+++++

++ +

+

+++++++

+

++

++

+

+

+

++++

++

+

+ ++++++ +++

++

++++ ++++++++

++++++

++ +++++

++

+++++++ +++ ++ +++++ +++++++++++

++

+

+

++++

+++++

+ + ++++ ++

++++

+

+++

++

++

+++ + ++

+

+

++

+++++

++ +

+

+++++++

+

++

++

+

+

+

++++

++

+

+ ++++++ +++

++

++++ ++++++++

++++++

++ +++++

++

+++++++ +++ ++ +++++ +++++++++++

++

+

+

++++

+++++

+ + ++++ ++

++++

+

+++

++

++

+++ + ++

Inviscid Simulation+

Figure 4. Viscous mesh and pressure coefficient distributions on a NACA 0012 airfoil surface (M∞ = 0.8,Re = 10× 106, and zero angle of attack).

x/C

Y

0.04 0.06 0.08 0.1

0.03

0.04

0.05

0.06

0.07

0.08

0.09

x/C

Y

0.35 0.4 0.45 0.5 0.55 0.6 0.65

0

0.05

0.1

0.15

0.2

0.25

Figure 5. Velocity vectors at the airfoil leading edge and at the upper surface shock wave region for a NACA0012 at transonic flight condition (M∞ = 0.8, Re = 10× 106, and zero angle of attack).

pressure coefficient distributions on the airfoil upper surface. A C-type mesh is used, with 60 nodes insidethe boundary layer thickness in the nominally wall-normal direction. Other 70 nodes are used to discretize theinviscid region away from the boundary layer. The flight condition considered is M∞ = 0.796, Re = 12×106

and zero angle of attack. Moreover, Fig. 6 also compares the influence of the ak2 coefficient of the artificialdissipation operator on the steady results. The interested reader should refer to section III for further detailsof the artificial dissipation operator. Results are in good agreement with experimental AGARD data.36The ak2 coefficient influence in viscous simulations is not so determinant to good results. Doubling thecoefficients yields very little variation in the pressure coefficient results over the airfoil.

Supercritical airfoils have some distinguished characteristics. Supercritical airfoil families are based onthe concept of local supersonic flow with isentropic recompression, characterized by a large leading edgeradius, reduced curvature over the mid airfoil section in the upper surface, and substantial aft camber.40In the present work, some supercritical airfoils are considered for transonic test cases. Slotted surfaces insupercritical airfoils are not introduce in the simulations, once this is out of the scope of the present work.The first supercritical airfoil considered in the present work is the OAT 1510 airfoil. For this airfoil, the

14 of 30


x/C0 0.2 0.4 0.6 0.8 1

Experimental Data

x/C

-Cp

0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5

URANS ak2 = 0.50

-Cp

0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5

URANS ak2 = 0.25

x/C0.35 0.4 0.45 0.5 0.55 0.6

Experimental Data

x/C

-Cp

0.35 0.4 0.45 0.5 0.55 0.60

0.2

0.4

0.6

0.8

URANS ak2 = 0.50

-Cp

0.35 0.4 0.45 0.5 0.55 0.60

0.2

0.4

0.6

0.8

URANS ak2 = 0.25

Figure 6. Comparison of results for a NACA 64A010 airfoil and effect of ak2 artificial dissipation coefficient(M∞ = 0.796, Re = 12× 106, and zero angle of attack).

transonic test case used considers M∞ = 0.724, Re = 3×106, based on the airfoil chord length, and α0 = 1.15deg. Figure 7 compares simulated results with experimental data from Ref. 37. One can observe in Fig.

X

Y

0 0.2 0.4 0.6 0.8 1

-0.4

-0.2

0

0.2

0.4

x/C

-Cp

0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5

1

Experimental Data-----------------

-----------------

------------------------------------------------

----------------

---------------------------------------------------------------------------------------------------------

-----------------

------------------------------------------------------------------------------------------------

-----------------------------------

-------------------------------------------------------

0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5

1

URANS Simulation-xxxxxx

xxx x x x x x x x x

xx

x x xx xx xxxx

Literature - SAxx

xxxxx

xxxxxxxx x x x xxxxxxxxxxxxx x xxx xxx

x xxx xxx

Literature - SAx

Figure 7. Unstructured mesh and pressure coefficient distributions over an OAT 1510 supercritical airfoil(M∞ = 0.724, Re = 3× 106, and α0 = 1.15 deg.).

7 that the numerical shock wave is located dowstream of the actual experimental shock. The results ofRef. 3 are also used to compare the present formulation with some numerical data which has solved thisproblem using the same turbulence model. The numerical literature data seems to corroborate the presentcalculations with the SA model. However, small differences are observed when comparing the results fromboth formulations at the shock-wave region. The shock wave captured by Ref. 3 is stronger than the shockwave captured in the present formulation. In other words, the present implementation of the SA modeldoes not seem to correctly capture the transonic shock wave for this test case. It is important to emphasizethat the numerical results of Ref. 3 are obtained usiig a 3-D URANS formulation, and a different spatialdiscretization numerical scheme. Figure 8 shows Mach number contours around the airfoil. It is clear thatthe shock wave is of a lesser intensity than that encountered in conventional airfoils.

Another interesting aspect of the present turbulent simulations can be observed in Fig. 9. This figure

15 of 30


X

Y

0 0.5 1-0.5

0

0.5

1Mach

10.90.80.70.60.50.40.30.20.1

Figure 8. OAT 1510 supercritical airfoil turbulent results: Mach number contours (M∞ = 0.724, Re = 3 × 106,and α0 = 1.15 deg.).

shows eddy viscosity coefficient contours around the airfoil and in the wake region. It is clear from thefigures that larger values of the eddy viscosity coefficient only appear downstream of the shock wave. The

X

Y

0 1 2

0

0.5

1

1.5

visc_t

50045040035030025020015010050

X

Y

0.6 0.8 1

0

0.2

visc_t

50045040035030025020015010050

Figure 9. Turbulence generation over the OAT 1510 supercritical airfoil and trailing edge region (M∞ = 0.724,Re = 3× 106, and α0 = 1.15 deg.).

results indicate good agreement with the experimental data. However, shock wave position is not as wellpredicted in the numerical simulations for this case, compared for instance with results for a conventionalairfoil. Figure 10 indicates that, in the leading edge region, the boundary layer is very thin and the velocityprofile variation is very abrupt in the normal direction. Moreover, the vector plots at the shock wave regionindicate that the shock wave–boundary interaction is not sufficiently strong to separate the flow field at thatregion, besides the presence of an adverse pressure gradient.

16 of 30


X

Y

0.03 0.035 0.04 0.045

0.03

0.035

0.04

0.045

X

Y

0.35 0.4 0.45

0.05

0.1

0.15

Figure 10. Velocity vector plots at leading edge and at shock wave region (M∞ = 0.724, Re = 3 × 106, andα0 = 1.15 deg.).

B. Study of the Proposed CFL Variation Correlation

The work of Strauss2 discussed the importance of some multigrid parameters. Such work studied the influenceof the number of meshes in the convergence rate, the number of relaxation steps in the coarser meshes, thenumber of smoothing steps before transfering data between adjacent meshes, and the efficiency of “V” or “W”cycles. In the present work, however, the authors concentrate, as far as the multigrid procedure is concerned,in the discussion of the proposed CFL variation formulation, Eq. 24, and the agglomeration ratio betweenconsecutive meshes.

The study is performed considering viscous simulations of the flow over the conventional NACA 64A010airfoil at transonic flight conditions. Freestream parameters are M∞ = 0.8, Re = 12 × 106, based on theairfoil chord, and zero angle of attack. Table 1 presents the results concerning the maximum CLF numberwith which the code can be run, with the multigrid procedure on, with and without the proposed CFLvariation formulation. The results are showing that one can run the code with CFL = 1.75, with multigrid,as long as the proposed CFL variation is active. On the other hand, if the procedure is not active, the highestvalue of CFL for stability, with multigrid, is 0.50. This demonstrates a clear advantage of the proposed CFLvariation formulation. Further evaluation of the effect of the proposed algorithm can be seen in Fig. 11.

Table 1. Effect of CFL variation formulation on the stability of simulations with multigrid.

Case CFL Using CFL Variation Not Using CFL Variation

1 1.75 Stable Unstable2 1.50 Stable Unstable3 1.25 Stable Unstable4 1.00 Stable Unstable5 0.50 Stable Stable

The left-hand side plot in Fig. 11 shows the results obtained in terms of convergence curves for cases 1 and5, as defined in Table 1, and obviously only for the stable cases. The authors emphasize that, of the threesimulation results shown in Fig. 11, two cases correspond to simulations performed with the CFL variationalgorithm turned on, with CFL = 0.50 and 1.75, and one test case was run without such algorithm, and usingCFL = 0.50. The right-hand side plot in Fig. 11 compares the resulting pressure coefficient distributionsobtained for the three simulations with the corresponding experimental data. The results demonstrate amuch faster convergence rate for case 1. Furthermore, for the two simulations which make up test case 5,

17 of 30


niter

log1

0(rh

smax

)

0 2000 4000 60000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

CFL=0.50 - No CFL Variation

niter

log1

0(rh

smax

)

0 2000 4000 60000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

CFL=0.50 - Use CFL Variation

niter

log1

0(rh

smax

)

0 2000 4000 60000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5


xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

xxxxxxxxxxxxxxxx

xxxxxxxxx

xxxxxxxxxx

xxx

xx

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

xxxxxxxxxxxxx

X

-Cp

0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5

CFL=1.75 - Use CFL Variationx

ooo

oo o o o

o o

o

oo o

o o o o o

Experimental Datao xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

xxxxxxxxxxxxxxxxxx

xxxxxxxxxxxxx

xxxxxxxx

xxxxxxxxx

xxx

xxx

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

X

-Cp

0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5

CFL=0.5 - No CFL Variationx

X

-Cp

0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5


Figure 11. CFL variation algorithm effects on the convergence rate and on the pressure coefficient distributionsfor a NACA 64A010 airfoil at M∞ = 0.8, α = 0 and Re = 12× 106.

there is a slight advantage in convergence rate for the case which uses the CFL variation algorithm. As oneshould expect, there is no relevant difference in the pressure coefficient distribution curves, regardless of theconvergence history. Moreover, the comparison with the experimental data is very good for this test case.

C. Unsteady Inviscid Results

An important aspect, when analyzing unsteady results, is a study of the aerodynamic hysteresis curves.These curves relate how fast the flow field responds with respect to the airfoil motion. The simulationsconsider a forced harmonic oscillatory excitation of the degrees of freedom of the structural model, and oneis interested in measuring the aerodynamic response of the flow field to such excitation. Typical resultsindicate the behavior of the aerodynamic force and moment coefficients as a function of motion of thestructural degrees of freedom. In the present work, the authors primarily consider pitch oscillations, i.e.,oscillations in angle of attack.

In order to start an unsteady simulation, one needs a converged steady solution for the mean angle ofattack of interest. The converged steady solutions obtained, and discussed in the previous subsection, areused to start the unsteady simulations. These steady simulations are considered converged when the lift andmoment coefficients stop to change as the iterations progress. It is important to emphasize that the initialsteady solutions must be fully converged in order to guarantee good behavior of unsteady calculations. Theunsteady simulations tend to blow up if convergence of the aerodynamic steady simulations has not beenachieved. The forced motion in the pitch degree of freedom is used as source of external harmonic excitation,at a prescribed frequency. As it is usual with aeroelastic applications, one typically refers to the reducedfrequency, which is defined as

κ =ω

U∞c

2. (58)

Here, c is the airfoil chord, U∞ is the free stream velocity magnitude and ω is the prescribed frequency ofpitch oscillation. Typically, values for the reduced frequency are lower than unity for all simulations, becausethis is the relevant range for aeroelastic applications. The prescribed airfoil pitch motion is performed as anincrement in angle of attack over the steady, mean angle of attack. In the present work, the pitch oscillationsare performed around the elastic axis. The instantaneous angle of attack can be written as

α = α0 + Δα sin (ωt) , (59)

where t is the time, α0 is the initial (steady) flow mean angle of attack and Δα is the small pitch oscillationamplitude. A typical oscillatory airfoil motion in pitch, with values in degrees, can be seem in Fig. 12.

The first unsteady test case considers a NACA 0012 airfoil, forced to oscillate in pitch with a reducedfrequency of 0.0814, mean angle of attack α0 = 0.016 deg., and an oscillation amplitude of 2.51 deg. The

18 of 30


Discrete Admensional Time

Pitc

h

0 50 100 150 200 250 300

-1

-0.5

0

0.5

1

Figure 12. Typical forced oscillation in pitch.

freestream Mach number for this test case is 0.755. The elastic axis is located at the quarter-chord point,measured from the leading edge. Figure 13 shows that there is a reasonable agreement between numericaland experimental data for lift and moment coefficients. Actually, although not shown in the figure, theagreement is as good in this case as it was obtained in previous simulations of the same problem.5,7, 14 The

AGARD

Pitch

Cl

-3 -2 -1 0 1 2 3

-0.4

-0.2

0

0.2

0.4

Euler

-2 -1 0 1 2-0.015

-0.01

-0.005

0

0.005

0.01

0.015

AGARD

Pitch

Cm

-2 -1 0 1 2-0.015

-0.01

-0.005

0

0.005

0.01

0.015

Euler

Figure 13. Unsteady lift and moment hysteresis curves for a NACA 0012 airfoil (M∞ = 0.755, κ = 0.0814,α0 = 0.016 deg. and Δα = 2.51 deg.).

experimental data was obtained from AGARD.32A second conventional airfoil, NACA 64A010, is also tested. This airfoil is thinner than the NACA

0012 airfoil and, therefore, the transonic shock wave over the airfoil is weaker than the one obtained for theNACA 0012 airfoil for the same flight condition. Figure 14 presents the results of the unsteady simulationfor the NACA 64A010 airfoil in pitch oscillation at M∞ = 0.80, α0 = 0 deg. and Δα = 1.01 deg. Thereduced frequency is 0.202. Moreover, the airfoil elastic axis is located at the quarter-chord point. Theresults are compared to experimental data from AGARD.33 It is clear that the lift coefficient hysteresisis adequately captured by the present simulations, whereas rather large differences occur in the moment

19 of 30


coefficient hysteresis curve.

Pitch

Cl

-1 -0.5 0 0.5 1

-0.1

-0.05

0

0.05

0.1Euler

-1 -0.5 0 0.5 1

-0.1

-0.05

0

0.05

0.1

AGARD

-1 -0.5 0 0.5 1-0.015

-0.01

-0.005

0

0.005

0.01

0.015

AGARD

Pitch

Cm

-1 -0.5 0 0.5 1-0.015

-0.01

-0.005

0

0.005

0.01

0.015Euler

Figure 14. Unsteady lift and moment coefficient hysteresis curves for a NACA 64A010 airfoil (M∞ = 0.80,κ = 0.202, α0 = 0 deg. and Δα = 1.01 deg.).

D. Dual Time Step Algorithm Performance

Unsteady simulations presented in the previous section did not use the dual time step (DTS) algorithm. Inother words, the number of inner iterations was equal to zero. A study of the DTS algorithm is performedbased on its main parameters, which are number of inner iterations and the real time step value. Allsimulations are performed for a NACA 0012 airfoil at M∞ = 0.755, α0 = 0.016 deg. and Δα = 2.51 deg.Reduced frequency of pitching oscillation is κ = 0.0814, based on Eq. 58 formulation. Method performanceis evaluated in terms of the hysteresis curves for the lift coefficient.

Figure 15 refers to unsteady inviscid results for flow simulation over a NACA 0012 airfoil at the flightcondition previously indicated. This test case does not use the DTS algorithm, but it evaluates the effectsof the real time step in the hysteresis curves for the lift coefficient. The results are indicating that, as onereduces the time step, the numerical results approach the experimental data. The figure shows numericalresults for real time steps of 0.01, 0.0015 and 0.0005 s. Clearly, the unsteady C� response for Δt = 0.01 s isvery poor. Results for the other two cases are similar, but there is a slight advantage for the Δt = 0.0005 scase. Probably the best conclusion that one can take from this exercise is that, in this particular test case,one needs at most a Δt = 0.0015 s, in order to obtain reasonable results. It must also be stated that, evenfor the smallest real Δt, the agreement with the experimental data is not perfect. However, the comparisonis somewhat equivalent to what has been obtained in the other results calculated in the present effort.

The number of inner iterations is also an important parameter to evaluate the performance of the DTSmethod. This parameter refers to the number of steady iterations to be performed within each real timestep of the simulation. As the number of inner iterations increases, a better converged result is obtainedwithin each time step. Obviously, this has a cost, and one has to weight the advantages and disadvantagesof increasing the number of inner iterations. The authors observe that the inner iterations are performedwith an explicit time marching scheme in the present approach. Figure 16 presents the results, for the sametest case, but using a real time step Δt = 0.01 s, and varying the number of inner iterations. Again, asone should expect, as the number of inner iterations increases, the hysteresis curve approaches the resultsobtained previously with the smallest real time step. One can observe that there is already a large differencebetween the results for no use of the DTS algorithm and its use with only 1 inner iteration. However, asone increases, initially, the number of inner iterations, consider for instance the curve for 5 inner iterationsin Fig. 16, the results approach the curve for no DTS algorithm use. This is probabely associated with theorder of accuracy of the implicit integration in real time and the explicit scheme used for the integration inpseudo time. However, as one keeps increasing the number of inner iterations, and hence allowing for trueconvergence within each real time step, the results tend to those obtained in the previous test case with the

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pitch

Cl

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Experimental Data

dt=0.0015dt=0.0005

pitch

Cl

-2 -1 0 1 2-0.6

-0.4

-0.2

0

0.2

0.4

0.6

dt=0.01

Figure 15. Influence of real time step in lift coefficient hysteresis curves (NACA 0012 airfoil, M∞ = 0.755,κ = 0.0814, α0 = 0.016 deg. and Δα = 2.51 deg.).

smallest time step, as already discussed.

E. Unsteady Viscous Results

This section discusses the influence of the viscous terms on the airfoil unsteady aerodynamic responses,considering a harmonic excitation in the structural degrees of freedom. As before, only harmonic motion inthe pitch degree of freedom is considered. The selected test case is the conventional NACA 64A010 airfoil attransonic speeds. The flow parameters are M∞ = 0.8, Re = 12×106 and zero mean (steady) angle of attack.The present calculations enforce the requirements of having, at least, 60 computational volumes within theboundary layer and of maintaining a value of y+ = 1 for the centroid of the first control volume off thewall. The Spalart-Allmaras turbulence model is used for turbulence closure in the present case. The farfield(outer) boundary is located at 50 chords away from the airfoil mid-chord point. The reduced frequency is0.202 and the pitching motion occurs about the quarter-chord point.

Figure 17 presents the hysteresis curves for lift and moment coefficients, due to the pitching oscillationof the configuration. In each plot, there are numerical results for two test cases, namely, without the useof the DTS algorithm, and with its use and considering 30 inner iterations within each real time step. Thecomputational results are compared to experimental data from AGARD.36 For the simulation without theuse of the DTS algorithm, the real time step is equal to 1.0 × 10−7 s. This is an extremely small timestep. On the other hand, for the simulation which takes advantage of the DTS algorithm, the real timestep is equal to 2.0 × 10−3 s, and 30 inner iterations are considered in the present case. With this latterreal time step, the angular displacement of the airfoil in angle of attack, during each real time step, is equalto 6.64 × 10−4 deg. Furthermore, for the sub-iterations which are performed using the explicit time marchscheme, the (pseudo) time step is equal to the real time step for the case without the DTS algorithm.

The results for the C� coefficient demonstrate that the viscous hysteresis curve, for the case which usesthe DTS algorithm, has better agreement with experimental data than the result calculated without the useof the DTS algorithm, despite the extremely small real time in this latter case. However, even for this betterresult, there is clearly room for improvement. Although the average slope of the hysteresis curve is quitesimilar to the experimental data, the hysteresis effect is not as pronounced in the numerical result as it isin the experimental case. The calculation without the use of the DTS algorithm yields a completely wronghysteresis behavior for the lift coefficient. On the other hand, the opposite trend is observed in the resultsfor the Cm hysteresis. Actually, the results without the use of the DTS algorithm do not agree very well

21 of 30


Experimental Data

pitch

Cl

-2 -1 0 1 2-0.6

-0.4

-0.2

0

0.2

0.4

0.6

No Inner Iter.1 Inner Iter.

Cl

-2 -1 0 1 2-0.6

-0.4

-0.2

0

0.2

0.4

0.6

5 Inner Iter.

Cl

-2 -1 0 1 2-0.6

-0.4

-0.2

0

0.2

0.4

0.6

10 Inner Iter.

Cl

-2 -1 0 1 2-0.6

-0.4

-0.2

0

0.2

0.4

0.6

25 Inner Iter.

Figure 16. Influence of the number of inner iterations in the unsteady lift airfoil response (NACA 0012 airfoil,M∞ = 0.755, κ = 0.0814, α0 = 0.016 deg. and Δα = 2.51 deg.).

-1 -0.5 0 0.5 1-0.15

-0.1

-0.05

0

0.05

0.1

0.15

AGARD R702

pitch

Cl

-1 -0.5 0 0.5 1-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Viscous 30 Inner Iter.

pitch

Cl

-1 -0.5 0 0.5 1-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Viscous No Dual

pitch

Cm

-1 -0.5 0 0.5 1-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Viscous 30 Inner Iter.

pitch

Cm

-1 -0.5 0 0.5 1-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Viscous No Dual

-1 -0.5 0 0.5 1-0.15

-0.1

-0.05

0

0.05

0.1

0.15

AGARD R702

Figure 17. Hysteresis curves for lift and moment coefficients including viscous terms (NACA 64A010 airfoil,M∞ = 0.8, Re = 12× 106, κ = 0.202, α0 = 0, and Δα = 1.01 deg.).

with the experimental data. However, this comparison becomes even worse when considering the numericaldata with the use of the DTS algorithm.

In an attempt to understand such differences, particularly in the moment coefficient curves, the authorsalso looked at the instantaneous pressure coefficient distributions on the airfoil throughout the oscillationcycle. Some of these results are shown in Fig. 18, which presents some instantaneous pressure distributions onupper and lower airfoil surfaces. The experimental AGARD data36 is also included in this figure. The resultsare clearly indicating that there are differences, even in the pressure coefficient distributions, between thenumerical results and the experimental data, regardless of the time march algorithm used for the simulations.Moreover, these differences occur independently of the phase angle considered along the oscillation cycle. Inparticular, it seems, from the results in Fig. 18, that the range of the displacement of the shock position in

22 of 30


xx xx

x x xx

x x

xx

xx x x x x

x/C

-Cp

0 0.2 0.4 0.6 0.8 1-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Exp. - Upper Surf.x

Phase = 60.0degs. - alpha = 0.875degs.

DUAL - 30 Inner Iter.NO DUAL

x/C

-Cp

0 0.2 0.4 0.6 0.8 1-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Exp. - Lower Surf.

xx xx

x x xx x x

xx

xx x x

x

x/C

-Cp

0 0.2 0.4 0.6 0.8 1-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Exp. - Upper Surf.x

Phase = 90.0degs. - alpha = 1.01deg.


x/C

-Cp

0 0.2 0.4 0.6 0.8 1-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Exp. - Lower Surf.

xx xx

x x xx

x x

x x xx x x x x

x/C

-Cp

0 0.2 0.4 0.6 0.8 1-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Exp. - Upper Surf.x

Phase = 180.0degs. - alpha = 0.00deg.


x/C

-Cp

0 0.2 0.4 0.6 0.8 1-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Exp. - Lower Surf.

xxx

xx x x

x x

xx x

xx x x x x

x/C

-Cp

0 0.2 0.4 0.6 0.8 1-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Exp. - Upper Surf.x

Phase = 270.0degs. - alpha = -1.01deg.


x/C

-Cp

0 0.2 0.4 0.6 0.8 1-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Exp. - Lower Surf.

Figure 18. Instantaneous pressure coefficient distributions on the NACA 64A010 airfoil (M∞ = 0.8, Re = 12×106,κ = 0.202, α0 = 0, and Δα = 1.01 deg.)

the numerical calculations is mucg smaller than the one observed for the experimental data, as one considersa full cycle of oscillation. This behavior can certainly explain the less pronounced hysteresis effect seen in thenumerical calculations. The results also seem to indicate that the numerical shock captured by the solutionalgorithm without the DTS scheme is typically downstream of the numerical shock captured with the DTSalgorithm. Finally, the numerical results with the DTS algorithm seem, in general, to be slightly closer tothe experimental data, in the 4 snapshots of the pressure coefficient distribution behavior shown in Fig.18, than the results without the DTS algorithm. However, this may simply be an impression from those 4figures, and the moment coefficient hysteresis curves are certainly stating otherwise.

XI. Aeroelastic Stability Analysis

The flutter analysis is performed for the transonic test case of a NACA 0012 conventional airfoil. Theflight conditions are M∞ = 0.8, Reynolds number based on the airfoil chord is equal to 10 × 106 and zeromean (steady) angle of attack. The structural parameters for the present test case are obtained from Ref.15These structural parameters are ah = −2.0, xα = 1.8, rα = 1.865, μ = 60, ωα = 100 rad/s, ωh = 100 rad/s.Moreover, ωτ , which is the reference circular frequency, is equal to the uncoupled frequency in the pitchingmode, i.e., ωτ = ωα. For the present flutter analysis, the characteristic dynamic pressure, q� varies from

23 of 30


0 to 1, with Δq� = 0.1. Moreover, the characteristic dynamic pressure is defined as (U�)2/μ. In order to

build the polynomial approximations which represent the aerodynamic transfer functions for the aeroelasticsystem, a small perturbation in both DOFs is performed separately. For the pitching DOF of the typicalsection model, a perturbation of 1× 10−5 deg. is used, and, for the plunging mode, the perturbation is equalto −1× 10−7 c. The real time step to integrate the system in time is equal to 1× 10−3 dimensionless unitsand the total integration time is approximately 30 dimensionless units. Moreover, 5 poles are consideredin the polynomial representation of the aerodynamic coefficients. In other words, 5 aerodynamic states areincluded in the aeroelastic system when considering the augmented state space equations.18

The stability analysis is performed by the construction of the aeroelastic stability root locus. The rootlocus is constructed as a function of the characteristic dynamic pressure and it allows the precise determina-tion of the flight speed in which the flutter instability occurs. Hence, system design can be performed basedon such results. The reduced frequency range of zero to one is considered. In the forthcoming discussion,the reasons for such choice are clarified. The available literature data are obtained from a linear theorysolution. Such results are used to compare both viscous and inviscid formulations in term of system stabilityboundaries. Finally, one can rewrite the expression for the reduced frequency as

κ =ωb

U∞=

1

U�

ω

ωα. (60)

Equation 60 demonstrates the importance of the characteristic speed in the numerical formulation of theproblem. The results for the aeroelastic stability root loci are shown in Fig. 19, considering linear theory(literature) data, inviscid and viscous calculations. These results are demonstrating a quite different behavior

xxxxx x x x x x x

ooooooooooo

-0.25 -0.2 -0.15 -0.1 -0.05 0 0.050

1

2

3

4

5

6

Plunge - ViscousPitch - Viscous

xo

Linear Theory

-0.25 -0.2 -0.15 -0.1 -0.05 0 0.050

1

2

3

4

5

6

Plunge - InviscidPitch - Inviscidω

/ωα

σ/ωα

Figure 19. Aeroelastic stability root locus for a NACA 0012 typical section model (M∞ = 0.8, Re = 10 × 106

and α0 = 0).

of the aeroelastic system for the case in which a viscous formulation is used for the aerodynamic operator.The root locus results indicate a very good agreement between inviscid18 and linear theories.15 Both DOFsconsidered in the simulations present the same behavior and the instability occurs in the pitch DOF in theq� range of 0.4 to 0.5. However, the results with the viscous formulation do not agree with data from eitherthe inviscid calculations or linear theory. The first point to be reported is that the viscous system is stable

24 of 30


in both pitch and plunge DOFs. Instability does no occur at any q� value. The curve which representsthe pith DOF clearly demonstrates that the direction of the root locus branch is preserved as q� increases.This behavior does no occur with the inviscid system, where the branch of the root locus for the pitch DOFchanges its direction and moves into the region of positive values for the real part of the flutter eigenvalues,which indicates unstable behavior and, hence occurrence of flutter.

It must be acknowledged that the results obtained in this test case were unexpected. Actually, for a NACA0012 airfoil at M∞ = 0.8, one would expect that the inclusion, or not, of the viscous terms in the aerodynamicformulation would not make that much of a difference, since the boundary layer should be quite energetic atthe foot of the shock. Hence, it was expected that the shock wave–boundary layer interaction would play alesser role in the mechanism for the aeroelastic instability. However, in an attempt to understand the previousaeroelastic results, it is important to analyze the unsteady aerodynamic responses for both DOFs, pitch andplunge. It is shown in Fig. 20 that viscous and inviscid unsteady aerodynamic responses, to an exponentially-shaped pulse18 in each DOF, are quite different. The viscous results demonstrate large oscillatory behavior

Time

Cl

0 5 10 15 20 25 30-0.0004

-0.0003

-0.0002

-0.0001

0

0.0001

0.0002

Cl - Pitch - Viscous

Cl

0 5 10 15 20 25 30-0.0004

-0.0003

-0.0002

-0.0001

0

0.0001

0.0002

Cl - Pitch - Inviscid

Time

Cm

0 5 10 15 20 25 30

-0.0002

-0.0001

0

0.0001

0.0002

0.0003

Cm - Pitch - Viscous

0 5 10 15 20 25 30

-0.0002

-0.0001

0

0.0001

0.0002

0.0003

Cm - Pitch - Inviscid

Time

Cl

0 5 10 15 20 25 30-0.003

-0.002

-0.001

0

0.001

0.002

Cl - Plunge - Viscous

Cl

0 5 10 15 20 25 30-0.003

-0.002

-0.001

0

0.001

0.002

Cl - Plunge - Inviscid

Time

Cm

0 5 10 15 20 25 30

-0.001

0

0.001

0.002

Cm - Plunge - Viscous

0 5 10 15 20 25 30

-0.001

0

0.001

0.002

Cm - Plunge - Inviscid

Figure 20. Lift and moment aerodynamic coefficients variation in time after application of an exponentially-shaped pulse excitation in each structural DOF (M∞ = 0.8, Re = 10× 106, α0 = 0 and Tmax = 1).

after 1 dimensionless time unit, which is the end of the exponentially-shaped pulse excitation (Tmax = 1), asindicated in Fig. 20. This plot compares the inviscid and viscous unsteady computational results in termsof the time history of the lift and moment coefficients for exponentially-shaped pulse excitation in eachstructural DOF. The airfoil excitation in pitching and plunging is performed for 0 ≤ t ≤ 1, which is theduration of application of the exponentially-shaped pulse function. Hence, after the excitation stops, the

25 of 30


aerodynamic response for the inviscid case essentially goes to zero very quickly. The same is not true for theviscous results, which behave as an oscillatory decaying system with quite large oscillations.

Another form of attempting to understand the previous aeroelastic result is to look at the lift and momentcoefficients in the Laplace domain, because this is the data used to construct the aerodynamic states.18Hence, Fig. 21 presents the Fast Fourier Transforms (FFT) of the unsteady aerodynamic responses, in termsof lift and moment coefficients, due to the exponentially-shaped pulse excitation in the pitch and plungemodes previously described. In other words, Fig. 21 has the FFT of the time histories presented in Fig.20, respectively. Moreover, the results from linear therory15 are also included in Fig. 21 for comparisonpurposes. This figure only shows the aerodynamic coefficients in the reduced frequency range 0 ≤ κ ≤ 1.Actually, in order to be more precise and since the perturbation in each DOF is not an impulse, but rather

0 0.2 0.4 0.6 0.8 1-6

-4

-2

0

2

4

6

8

10

12

14

16

Linear Theory

k

Re(

Cl),

Im(C

l)

0 0.2 0.4 0.6 0.8 1-6

-4

-2

0

2

4

6

8

10

12

14

16

Re(Cl) - InviscidIm(Cl) - Inviscid

k

Re(

Cl),

Im(C

l)

0 0.2 0.4 0.6 0.8 1-6

-4

-2

0

2

4

6

8

10

12

14

16

0 0.2 0.4 0.6 0.8 1-6

-4

-2

0

2

4

6

8

10

12

14

16

x

x xx

xx

x

o o o oo

o

o

k

Re(

Cl),

Im(C

l)

0 0.2 0.4 0.6 0.8 1-6

-4

-2

0

2

4

6

8

10

12

14

16

Re(Cl) - ViscousIm(Cl) - Viscous

xo

Real

Imaginary

0 0.2 0.4 0.6 0.8 1

-4

-2

0

2

4

Linear Theory

Imaginary

Real

k

Re(

Cm

),Im

(Cm

)

0 0.2 0.4 0.6 0.8 1

-4

-2

0

2

4 Re(Cm) - InviscidIm(Cm) - Inviscid

0 0.2 0.4 0.6 0.8 1

-4

-2

0

2

4

x

x x

x

x

x

x

oo o o

o

o

o

0 0.2 0.4 0.6 0.8 1

-4

-2

0

2

4

Re(Cm) - ViscousIm(Cm) - Viscous

xo

0 0.2 0.4 0.6 0.8 1-6

-4

-2

0

2

4

6

8

10

12

14

16

Linear Theory

k

Re(

Cl),

Im(C

l)

0 0.2 0.4 0.6 0.8 1-6

-4

-2

0

2

4

6

8

10

12

14

16

Re(Cl) - InviscidIm(Cl) - Inviscid

k

Re(

Cl),

Im(C

l)

0 0.2 0.4 0.6 0.8 1-6

-4

-2

0

2

4

6

8

10

12

14

16

0 0.2 0.4 0.6 0.8 1-6

-4

-2

0

2

4

6

8

10

12

14

16

xx x x

x xx

oo

oo

oo

o

k

Re(

Cl),

Im(C

l)

0 0.2 0.4 0.6 0.8 1-6

-4

-2

0

2

4

6

8

10

12

14

16


xo

Imaginary

Real

0 0.2 0.4 0.6 0.8 1

-4

-2

0

2

4

6

8

Linear Theory

Imaginary

Real

k

Re(

Cm

),Im

(Cm

)

0 0.2 0.4 0.6 0.8 1

-4

-2

0

2

4

6

8

Re(Cm) - InviscidIm(Cm) - Inviscid

0 0.2 0.4 0.6 0.8 1

-4

-2

0

2

4

6

8

xx

x xx

x

x

oo

o

oo

o0 0.2 0.4 0.6 0.8 1

-4

-2

0

2

4

6

8


xo

Figure 21. Real and imaginary parts of lift and moment aerodynamic coefficients due to an exponentially-shaped pulse excitation in each structural DOF (M∞ = 0.8, Re = 10× 106, α0 = 0 and Tmax = 1).

a smooth exponentially-shaped pulse, the data in Fig. 21 represent the result of the FFT of the data inFig. 20 divided by the FFT of the corresponding exponentially-shaped pulse excitation. In other words, thisdata represents the aerodynamic transfer functions for the DOF’s considered for the typical section model.The results indicate very good agreement between the inviscid formulation and linear theory, for both realand imaginary parts of the aerodynamic coefficient transfer functions. However, the viscous formulationresults do not agree as well with the rest of the data. The discrepancies are, actually, quite large, whichthen explains why the aeroelastic stability analysis using the viscous data has yielded very different results.The agreement of the viscous results becomes even worse if one extends the range of reduced frequencies of

26 of 30


interest. This is the main reason why the aeroelastic analysis used aerodynamic states interpolated from thedata only in the 0 ≤ κ ≤ 1 reduced frequency range.

This last comment prompts the consideration of how well the aerodynamic coefficient transfer functiondata can be approximated by the aerodynamic interpolating polynomials. This issue is addressed in theresults presented in Fig. 22, which compares the original unsteady aerodynamic coefficient transfer functiondata, obtained from the FFT of the time histories from the viscous calculations, with the results obtainedfrom the interpolating polynomials which approximate such data. The results are, again, presented in termsof C� and Cm transfer functions for both pitch and plunge DOFs, respectively. It is clear from the figure

Reduced Frequency

Re(

Cl)

&Im

(Cl)

-Pitc

h

0 0.2 0.4 0.6 0.8 1 1.2-6

-4

-2

0

2

4

6

8 Re(Cl) - InterpolationIm(Cl) - Interpolation

x

x xx

x

x

xx

o o o oo

o

oo

0 0.2 0.4 0.6 0.8 1 1.2-6

-4

-2

0

2

4

6

8


xo

Reduced Frequency

Re(

Cm

)&Im

(Cm

)-P

itch

0 0.2 0.4 0.6 0.8 1 1.2-6

-4

-2

0

2

4

6

8Re(Cm) - InterpolationIm(Cm) - Interpolation

x

x xx

x

xx

x

oo o o

o

o

oo

0 0.2 0.4 0.6 0.8 1 1.2-6

-4

-2

0

2

4

6

8


xo

Reduced Frequency

Re(

Cl)

&Im

(Cl)

-Plu

nge

0 0.2 0.4 0.6 0.8 1 1.2-10

-8

-6

-4

-2

0

2

4

6

8

Re(Cl) - InterpolationIm(Cl) - Interpolationx

xx x

xx

xx

oo

oo

o

oo

o

0 0.2 0.4 0.6 0.8 1 1.2-10

-8

-6

-4

-2

0

2

4

6

8


xo

xx

x xx

xx

x

oo

oo

o

oo

o0 0.2 0.4 0.6 0.8 1 1.2

-6

-4

-2

0

2

4

6


xo

Reduced Frequency

Re(

Cm

)&Im

(Cm

)-P

lung

e

0 0.2 0.4 0.6 0.8 1 1.2

-6

-4

-2

0

2

4

6

Re(Cm) - InterpolationIm(Cm) - Interpolation

Figure 22. Comparison of original aerodynamic coefficient transfer function data and the results of the aero-dynamic interpolating polynomials (M∞ = 0.8, Re = 10× 106, α0 = 0, Tmax = 1 and 0 ≤ κ ≤ 1).

that, at least in the 0 ≤ κ ≤ 1 reduced frequency range, the agreement is very good. In this particularcase, the aerodynamic coefficient interpolating polynomials have considered 5 poles. The results would notbe much different if 10 aerodynamic states had been considered. On the other hand, if one increases thereduced frequency range of interest to 0 ≤ κ ≤ 3, even with 10 aerodynamic states, it would not be possibleto obtain interpolation polynomials that would adequately approximate the FFT data.

27 of 30


XII. Concluding Remarks

The paper has presented the development of an effort which had the objective of including viscous termsin a numerical tool which is used to generate the aerodynamic operator for aeroelastic stability analysis. Inthis regard, the overall formulation is presented and discussed in the paper. In this development process,several numerical aspects of the available computational tool had to be improved in order to allow unsteadyresults in a reasonable amount of time. Viscous flows are computed using the unsteady Reynolds-averagedNavier-Stokes equations in 2-D, and turbulence closure is achieved with the Spalart-Allmaras one-equationturbulence model. Moreover, the paper has presented steady aerodynamic results for three different airfoilconfigurations, namely the NACA 0012, NACA 64A010 and OAT 1510 airfoils, using both the inviscidand the turbulent viscous formulations available. Steady results, in general, yield good agreement with theavailable literature data. Therefore, it seems that the implementation of the various CFD aspects of theaerodynamic simulation code has been correctly executed.

Unsteady calculations, on the other hand, demonstrated to be quite problematic and certainly deservefurther attention. Viscous unsteady calculations have led to the implementation of a dual time step algorithm,which was absolutely essential to allow the calculations here discussed in a reasonable time frame. Otherwise,the use of the original explicit time stepping scheme would have yielded computational times which wereextremely high, because the maximum allowed stable time step for viscous unsteady simulations was verysmall. Aerodynamic hysteresis curves have been obtained for both viscous and inviscid calculations. Again, ingeneral, the present computational results for viscous simulations have shown agreement with the literaturedata which is comparable to the sort of agreement which the authors were used to obtain with inviscidcalculations. Moreover, typically, the use of the dual time step algorithm has contributed to improve thequality of the unsteady results, as well as to make such computations more affordable. Both NACA 0012and NACA 64A010 airfoil configurations have been considered for such forced harmonic oscillation studies.

Aeroelastic stability analyses were performed for a NACA 0012 airfoil-based typical section model, withplunge and pitch degrees of freedom. The aerodynamic data, to create the augmented aerodynamic states,was generated by the unsteady CFD tool using both inviscid and viscous simulations. The results obtainedwith the inviscid formulation exactly reproduce previous studies by some of the present authors and theyalso agree very well with independent data in the literature. The results obtained with the use of a vis-cous formulation yielded, however, a quite different aeroelastic behavior. The several studies performedindicated that the unsteady aerodynamic responses, in terms of lift and moment coefficient time histories,to exponentially-shaped pulses in the two structural degress of freedom are completely different when theviscous terms are included in the formulation. Such unsteady aerodynamic responses are the raw CFD datafrom which one is eventually able to construct aerodynamic transfer functions in the frquency domain and,hence, the augmented aerodynamic states for aeroelastic analyses. Therefore, it is no surprise that the aeroe-lastic stability behavior of the system is completely different for the calculations performed using a viscousformulation.

At the present time, the authors have no plausible explanation for the very different unsteady aerodynamicbehavior in the pulse responses, especially because the harmonic forced oscillation responses did not indicateany fundamentally different behavior for the simulations with the viscous terms included in the formulation.Hence, the authors have to admit that such totally different results were unexpected, at least for a NACA0012 airfoil configuration. Therefore, in summary, the present effort should probably best be seen as a workin progress, in the sense that such discrepant results certainly deserve further investigation.

Acknowledgments

The authors gratefully acknowledge the financial support for this research provided by Coordenação deAperfeiçamento de Pessoal de Ensino Superior, CAPES, which provided a MS scholarship to the first author.The authors would also like to acknowledge Conselho Nacional de Desenvolvimento Científico e Tecnológico,CNPq, which partially supported the project under the Research Grant No. 312064/2006-3. Further partialsupport was provided by Fundação de Amparo à Pesquisa do Estado de São Paulo), FAPESP, under ProcessNo. 2004/16064-9. Moreover, the authors want to gratefully acknowledge the very fruitful discussions withMr. Edson Basso, who provided interesting and constructive insights that helped shape the present work.

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