Free-Wake Modeling via Conformal Mappings

for the Aeroelastic Analysis of a Typical Section

Cristina Riso∗

“Sapienza” Universita di Roma, Rome, Italy, 00184

Nonlinear aeroelastic modeling is increasingly important for flexible or rapidly ma-neuvering aircraft design, sensitivity analysis, and optimization. Nevertheless, the largecomputational burden may make high-order nonlinear models not suitable for multiplesimulations. In order to minimize the computational effort and at the same time takeinto account the most relevant physical phenomena, reduced-order models are currentlyunder investigation. The purpose of this work is to present a free-wake aeroelastic modelfor a typical section, based on a novel method to compute the aerodynamic force andmoment on arbitrarily moving bodies in incompressible potential flows. The proposedapproach involves the Schwarz function of the body boundary and conformal mappings.The aerodynamic loads are analytically evaluated, with major advantages for engineeringinterpretation and computational efficiency, and coupled with the equations of motion toderive a simplified nonlinear aeroelastic model suitable for what-if analysis and prelimi-nary design. Numerical results are presented to demonstrate the capability of the modelto accurately predict the unsteady flow field features due to large-amplitude motions andquantify their effect on aeroelastic behavior.

I. Introduction

In the aircraft design process, aeroelastic analysis is typically carried out by means of linearized ap-proaches.1,2 The associated computational tools are well-established, available in general purpose commer-cial codes as well as in those developed in-house by aircraft manufacturers. However, an accurate predictionof the aeroelastic behavior of highly flexible or rapidly maneuvering aircraft configurations requires advancednonlinear modeling capabilities, due to the fact that the simplifying assumptions of linearized kinematicsand prescribed wake geometry are not reasonable. Unfortunately, high-fidelity nonlinear models are compu-tationally demanding. Therefore, they may not suit the early stages of the design cycle, sensitivity analysisand optimization, which involve multiple simulations. The development of low-order models is thus of pri-mary concern, as they allow to reduce the computational burden while still capturing the most relevantaerodynamic and structural nonlinearities.

Despite the great simplification over the reality of finite wings, typical section models1 are particularlyinteresting. The possibility to obtain closed-form solutions for the aerodynamic loads under the assumptionof small disturbances has made such models an important source of qualitative information concerningunsteady airfoil behavior and fixed-wing flutter. Wagner3 investigated the growth of lift on a thin airfoilin incompressible potential flow due to a step change in angle of attack. Kussner4 treated the lift responsedue to a vertical gust. Theodorsen5 presented the lift and pitching moment on a thin airfoil with a flapundergoing small harmonic oscillations in plunge and pitch, and he related the circulatory part of the flow toa function of the sole reduced frequency. His work also included a detailed investigation on the mechanismof flutter,5,6 which pointed out the role played by geometric, mass, and stiffness properties in the loss ofstability. The relevance of these results and their application in aeroelastic design of large aspect-ratio wings1

gives a primary motivation for removing the simplifying assumptions of small-amplitude motions and flatwake, so that typical section models can be used to better understand the physical mechanisms involved innonlinear aeroelastic behavior.

∗Currently Ph.D. Candidate, Dept. of Mechanical and Aerospace Engineering, Via Eudossiana 18; cristina.riso@uniroma1.it

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The present work is aimed to derive a semi-analytical, free-wake aeroelastic model for a typical sectionand numerically simulate its arbitrary motion when elastically connected to a rigid support. This is doneunder the physical hypotheses of inviscid and incompressible fluid in attached, planar, and potential flow.1

A general method to compute the aerodynamic force and moment acting on arbitrarily moving bodies isproposed, which involves the Schwarz function7 of the body boundary and the complex potential of theflow.8,9 Removing the assumption of small disturbances, the method is specialized to a typical section bymeans of a time-dependent conformal map.10 The wake released from the trailing edge is discretized in pointvortices, which move according to Biot-Savart law.11,12 The aerodynamic loads are evaluated via complexanalysis10 and coupled with the equations of motion to finally derive the aeroelastic model.

The proposed approach takes into account large-amplitude displacements and consequent wake patternswithout introducing further approximations. Moreover, it points out the dependency of aeroelastic behavioron asymptotic flow and typical section parameters, unsteady boundary conditions, and free-wake geometry,providing good physical insight and leading to relevant applications in aeroelastic design. Since the aerody-namic loads are analytically evaluated, the present model also results in an efficient numerical implementationthat enables to perform nonlinear aeroelastic analyses with moderate computational resources.

II. Aerodynamic Force and Moment on Arbitrarily Moving Bodies

In this section a novel method to compute the aerodynamic force and moment on arbitrarily moving bodiesin two-dimensional incompressible potential flows is presented. The method is based on the application ofthe Schwarz function7 of the body boundary. This leads to general analytical formulas for the aerodynamicloads in terms of a complex potential of the flow,8,9 which extend Blasius theorem8,9 to the case of unsteadyboundary conditions. Since no assumption is made on the section shape and velocity, the present formulationalso applies to in-plane elastic bodies.

A. Mathematical Preliminaries

Assume an inviscid and incompressible fluid in attached, planar, and potential flow. Identify the plane of theflow (x, z) with the complex plane, so that the position vector is written as the complex variable x = x+ i z.Since the velocity potential ϕ = ϕ(x, z; t) and stream function ψ = ψ(x, z; t) satisfy Laplace equation, thecomplex potential of the flow8,9

w(x; t) := ϕ(x, z; t) + i ψ(x, z; t) (1)

is analytic10 in the fluid domain. Denoted by u = u(x, z; t) and w = w(x, z; t) the horizontal and verticalvelocity components, respectively, so that the velocity vector is written as u(x; t) = u(x, z; t) + i w(x, z; t),the complex derivative10 of w gives the conjugate velocity,10 namely ∂xw = u.

B. Aerodynamic Force

Consider the body boundary ∂Ωb (counter-clockwise oriented) and its Schwarz function7 Φa. The unittangent and outward normal vectors are τ and n = −i τ , respectively. The curve element is dx = τds,whereas its conjugate can be recast as dx = dΦ = dx ∂xΦ. The body boundary is assumed to move with avelocity ub(x; t), which depends on the position x ∈ ∂Ωb and time. The analytic continuations10 of ub andub are also introduced, the latter obtained replacing x with the Schwarz function7 of the curve ∂Ωb.

The aerodynamic force (per unit span length) is

F a =

∫∂Ωb

ds (−pn) = −iρ∫∂Ωb

dx

(∂tϕ+

|u|2

2

), (2)

where p is pressure and ρ is the fluid density. Pressure is rewritten using Bernoulli theorem.The first contribution in Eq. (2) is handled using the relation ∂tϕ =

(∂tw + ∂tw

)/2 and rewriting the

curve element dx as the conjugate of dx = dΦ. Thus,∫∂Ωb

dx ∂tϕ =1

2

∫∂Ωb

dx(∂tw + ∂tw

)=

1

2

( ∫∂Ωb

dx ∂tw +

∫∂Ωb

dx ∂xΦ ∂tw

). (3)

aThe Schwarz function7 is such that Φ(x) := x, x ∈ ∂Ωb, whereas it is elsewhere defined by means of analytic continuation.10

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The second contribution in Eq. (2) is evaluated by observing that the differential dw = dϕ + i dψ has tobe expressed on the body contour in terms of the unsteady boundary condition, namely using the relationdψ = ds ∂sψ = ds u · n = ds ub · n = i (ub dx− ub dx)/2. Hence

dw = dϕ+1

2(ub dx− ub dx) , (4)

which givesb

1

2

∫∂Ωb

dx |u|2 =1

2

∫∂Ωb

dw ∂xw

=1

2

∫∂Ωb

dϕ ∂xw +1

4

∫∂Ωb

(ub dx− ub dx) ∂xw

=1

2

∫∂Ωb

[dw − 1

2(ub dx− ub dx)

]∂xw +

1

4

∫∂Ωb

(ub dx− ub dx) ∂xw

=1

2

( ∫∂Ωb

dw ∂xw +

∫∂Ωb

dx ub ∂xw −∫∂Ωb

dx ub ∂xw

)=

1

2

[ ∫∂Ωb

dx (∂xw)2

+

∫∂Ωb

dx ∂xΦ ub ∂xw −∫∂Ωb

dx ub ∂xw

]. (5)

Substituting Eqs. (3) and (5) into Eq. (2) the aerodynamic force can be finally expressed as

F a = −iρ2

[ ∫∂Ωb

dx ∂tw +

∫∂Ωb

dx ∂xΦ ∂tw +

+

∫∂Ωb

dx ∂xΦ ∂xw ub −∫∂Ωb

dx ∂xw ub +

∫∂Ωb

dx (∂xw)2

]. (6)

Note that the first two integrals in Eq. (6) are identical if the body does not movec (ub ≡ 0). Therefore,

F a = −iρ∫∂Ωb

dx ∂tw −iρ

2

∫∂Ωb

dx (∂xw)2. (7)

This result is known as the generalized form of Blasius theorem,9,13 which takes into account flow unsteadi-ness but does not include body motion. For steady flows (∂tw ≡ 0), Eq. (7) reduces to Blasius theorem.8,9

C. Aerodynamic Moment

Denoted by y the axis normal to the plane of the flow, the aerodynamic moment about the point x0 is

Ma(x0) =

∫∂Ωb

ds (x− x0)× (−p n)

y

=

∫∂Ωb

ds x× (−p n)

y

− x0 × F a

y

. (8)

The component of x0 × F a along y is evaluated as Im (F a x0), whereas the integral on the right-hand sideof Eq. (8) is the moment about the origin, denoted by Ma(0). This latter quantity is expressed as follows.

The component of x× n along y can be rewritten as Im(nΦ), so thatd

Ma(0) = +Im

[ ∫∂Ωb

ds (−p) n Φ

]= −Re

[ ∫∂Ωb

dx Φ (−p)]

= −ρ Re

[ ∫∂Ωb

dx Φ

(∂tϕ+

|u|2

2

)]. (9)

bSince it is real, the differential dϕ can be replaced with its conjugate. The latter is evaluated using Eq. (4).cFor stationary bodies, the stream function and its time derivative are constant (in space) on the body boundary. As a

consequence, the integrals of the imaginary part of ∂tw in Eq. (6) identically vanish, whereas those of the real part are equal.dThe arbitrary time-dependent contribution in Bernoulli theorem does not play any role in Eq. (9), due to the fact that the

integral of Φ is an imaginary complex number.7

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The two contributions in Eq. (9) are expressed following the same approach used in the derivation of Eqs. (3)and (5), but the procedure14 is omitted for the sake of conciseness. Since the imaginary parts of the integralsdo not play any role in Eq. (9), it finally achieves the form

Ma(0) = −ρ2

Re

[ ∫∂Ωb

dx Φ ∂tw +

∫∂Ωb

dx x ∂xΦ ∂tw +

+

∫∂Ωb

dx x ∂xΦ ∂xw ub −∫∂Ωb

dx x ∂xw ub +

∫∂Ωb

dx x (∂xw)2

]. (10)

The preceding result reduces to Blasius theorem8,9 for stationary bodies in steady flows (ub ≡ 0, ∂tw ≡ 0).Note that in deriving Eqs. (6) and (10) the body motion is taken into account by locally applying the

unsteady boundary condition u ·n = ub ·n. Therefore, the present formulation also applies to flexible bodies.Such a result could not have been achieved by assuming the body at rest in a body-fixed reference frameand taking into account its motion through the relative flow velocity.15 Indeed, this approach is evidentlylimited to rigid sections. Since the unsteady boundary condition gives an integral involving ub, which hasto be rewritten as analytic function, the application of the Schwarz function7 is thus essential.

III. Specialization to a Typical Section

The aerodynamic model is here specialized to a typical section, modeled as a rigid flat plate.1,5 Thecomplex potential of the flow8,9 is evaluated by means of a conformal map10 that transforms the plate plane(x-plane) onto an auxiliary plane (ζ-plane) in which the plate becomes a circle.9 Removing the assumptionof small disturbances, the usual Joukowski conformal transformation1,9 is replaced by a time-dependentmap that takes into account the plate arbitrary motion. The flow is characterized by enforcing the unsteadyboundary condition on the plate and Kutta condition at the trailing edge. The aerodynamic loads are finallyevaluated by means of Cauchy residue theorem10 and principal value integrals.10

A. Conformal Maps

The circle of radius R and center on the origin (ζ-plane) is mapped onto the flat plate (x-plane) of chord l,midchord point on H = H(t) and angle of attack α = α(t) according to the conformal maps

x(ζ; t) = H +l χ

4

(ζ

R+R

ζ

),

ζ(x; t) =2R

lχ[(x−H) +

√(x− x−) (x− x+)

],

(11)

where χ := exp(iα) and the points x± = H ± l/(2χ) are the plate edges. An example (R = 1m, l =2m, α = 45, H = Hx + i Hz, Hx = 2m, Hz = 1m) is illustrated in Fig. 1.

(a) ζ-plane (b) x-plane

Figure 1. Conformal transformation of a circle (ζ-plane) onto an inclined flat plate (x-plane).

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The plate intrinsic reference system τ = −χ and n = i χ is adopted, in which the position (H) andvelocity (H) of the midchord point have the following components:

Hτ := H · τ = −Re(χ H

)= −Re

(χ H

), Hn := H · n = +Re

(i χ H

)= −Re

(i χ H

),

Vτ := H · τ = −Re(χ H

)= −Re

(χ H

), Vn := H · n = +Re

(i χ H

)= −Re

(i χ H

).

(12)

The curve element dx is deduced from the map x(ζ; t), whose time derivative gives the body velocity ub.The conjugate quantities dx and ub are rewritten as analytic functions using the Schwarz function7 of thecircle Φ(ζ) = R2/ζ. The derivatives of the map ζ(x; t) are also evaluated from Eq. (11). The results are:

dx = χl

4R

ζ2 −R2

ζ2dζ , dx = dx ∂xΦ = χ

l

4R

ζ2 −R2

ζ2dζ ,

ub = H − i χ αl

4

(ζ

R+R

ζ

), ub = H + i χ

αl

4

(ζ

R+R

ζ

),

∂tζ = i α ζ +2R

l

i αlR ζ − 2 H χ ζ2

ζ2 −R2, ∂xζ = χ

4R

l

ζ2

ζ2 −R2.

(13)

Finally, the asymptotic velocity in the circle plane v∞ is expressed in terms of the one in the plate planeu∞ := u∞ exp(iβ) by observing the asymptotic behavior of the map x(ζ; t), which yields

v∞ =l

4Ru∞ χ =

lu∞4R

exp [i (α+ β)] . (14)

B. Complex Potential

The complex potential of the flow in the circle plane (w) is expressed as the sum of the contributions dueto the asymptotic flow (w∞), plate dynamics (wd), body circulation (wc), and wake (ww), namely,

w(ζ; t) = w∞(ζ; t) + wd(ζ; t) + wc(ζ; t) + ww(ζ; t) . (15)

In order to satisfy the unsteady boundary condition, the complex potential in the absence of wake isassumed of the forme

w∞(ζ; t) + wd(ζ; t) + wc(ζ; t) := v∞(t) ζ +A(t)

ζ+B(t)

ζ2 +C(t) log ζ . (16)

Deduced the normal velocities on the plate using Eqs. (13) and (16) and imposed ub · n = u · n, thetime-dependent coefficients turn out to be14

A = R2

(v∞ − i

lVn2R

), B = i α

R2l2

16, C =

Γb2πi

, (17)

where Γb is the circulation around the plate. Substituting Eq. (17) into Eq. (16) gives

w∞(ζ; t) = v∞ ζ + v∞R2

ζ, wd(ζ; t) =

lVn2i

R

ζ+ i α

l2

16

R2

ζ2 , wc(ζ; t) =Γb2πi

log ζ . (18)

The wake shed from the trailing edge is discretized in point vortices.11,12 Denoted by Γj and ζj = ζj(t)the circulation and position (ζ-plane) of the j-th point vortex, respectively, the last term in Eq. (15) can bedeveloped as9

ww(ζ; t) =

n∑j=1

Γj2πi

log [ζ − ζj ]− log

[ζ −R2/ζj

]+ log ζ

. (19)

The body circulation is related to the initial one and to the wake vorticity content by Kelvin theorem.

eIn the absence of wake, the complex potential w is analytic outside the circle |ζ| = R. As a consequence, it can be thereexpanded in Laurent series10 about the point ζ = 0. Once it is observed that regular terms of degree larger than 1 lead to avelocity that diverges as ζ → ∞, whereas singular terms of degree smaller than −2 lead to contributions as cos 2θ, cos 3θ, . . .in the normal velocity u · n on the plate contour, the truncated series in Eq. (16) follows.

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Vortex kinematics is given by Biot-Savart law and numerically simulated by means of a time-marchingprocedure. A semi-analytical aerodynamic model is thus derived, in which the assumption of flat wake iscompletely removed. Vortex shedding is modeled using the fixed-position method.12 At each time-step, avortex is released from the trailing edge. The position of appearance of the nascent vortex11,12 is assumedin the circle plane as ζ? = R(1 + δ), where δ 1. Its circulation Γ? is evaluated using the Kutta condition

∂ζ (w∞ + wd + w?c + w?

w)ζ=+R

+Γ?

2πi

1

R−R (1 + δ)− 1

R−R2/ [R (1 + δ)]+

1

R

= 0 , (20)

where the circulation (w?c ) and wake (w?

w) complex potentials do not take into account the nascent vortex.Equation (20) states that vortex shedding ensures v = 0 at the point ζ = +R, so that the conjugate velocityin the plate plane u = ∂ζw ∂xζ = v ∂xζ remains finite on the trailing edge x = x+.

Once the complex potential in the circle plane is known, the one in the plate plane, denoted by w, isevaluated via conformal mapping as

w(x; t) = w [ζ(x; t); t] . (21)

Hence, its derivatives are given by

∂tw = ∂tw + ∂ζw ∂tζ , ∂xw = ∂ζw ∂xζ . (22)

The preceding quantities are expressed using the derivatives of the map ζ(x; t), given by Eq. (22), and theones of the complex potential w, deduced from Eqs. (18) and (19). The details of the computations areomitted for the sake of conciseness.

C. Aerodynamic Loads

The aerodynamic loads on the typical section are obtained by specializing Eqs. (6) and (10) using therelations derived in the preceding subsection. The integrals are analytically evaluated by means of Cauchyresidue theorem.10 Because of the leading-edge singularity, Cauchy principal value integrals10 are also used.

Defined the (time-dependent) coefficients depending on the wake geometry and vorticity content

a(k) = a(k)r + i a

(k)i := Rk

n∑j=1

Γj

ζkj,

b∓ = b∓r + i b∓i := R

n∑j=1

Γjζj ±R

,

c∓ = c∓r + i c∓i := 2R2n∑

j,k=1

ΓjΓk

(ζj ±R)(ζk −R2/ζj),

d = dr + i di := 2R2n∑

j,k=1

ΓjΓk

ζj(ζk −R2/ζj

) ,

(23)

the normal and tangentialf components of the aerodynamic force (F a = F an n+ F a

τ τ ), divided by the fluiddensity, are written as

F an

ρ= −π

4l2 Vn +

[−u∞ cos (α+ β)− Vτ +

1

πl

(b−i + b+i

) ]Γb +

+π

4l2[u∞ sin (α+ β) + u∞

(α+ β

)cos (α+ β)

]− l

2

(a(1)r + α a

(1)i

)+ (24)

+1

2πl

(2b−r b

−i − 2b+r b

+i + c+i − c

−i

)+αl

4

(b−i + b+i

)+

(b+i − b

−i

)[u∞ sin (α+ β)− Vn

],

F aτ

ρ=

1

4πl

(Γ2b − 2q1Γb + q2

), (25)

fAccording to Eq. (2), it is expected to have F aτ = 0. Such condition is numerically satisfied.

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where

q1 =− πl[u∞ sin (α+ β)− Vn +

αl

4

]+ 2b+r ,

q2 = + π2l2[u∞ sin (α+ β)− Vn +

αl

4

]2

− 4πlb+r

[u∞ sin (α+ β)− Vn +

αl

4

]+

+ 2

(b− 2r + b− 2

i + b+ 2r − b+ 2

i + c+r − c−r).

(26)

The aerodynamic moment about the origin can be written as Ma(0) = −F anHτ + F a

τ Hn + Ma (H), wherethe moment about the midchord point, divided by the fluid density, is

Ma (H)

ρ= +

π

128l4 α− π

4l2 [u∞ sin (α+ β)− Vn] [u∞ cos (α+ β) + Vτ ] +

+l

2[u∞ cos (α+ β) + Vτ ] a(1)

r +l

2

(b−i + b+i − a

(1)i

)[u∞ sin (α+ β)− Vn] + (27)

− l2

16

(a(2)r + 2 α a

(2)i

)+

(Γb2π

+αl2

8

)(b+i − b

−i

)+

1

4π

(−2b−r b

−i − 2b+r b

+i + c+i + c−i − di

).

Equation (24) shows that the aerodynamic force has a contribution proportional to Vn, which also dependson the quantity H. This term leads to the identification of the virtual mass1 πl2ρ/4. Similarly, Eq. (27)shows that the virtual moment of inertia1 (about the midchord point) is πl4ρ/128. Finally, the aerodynamicloads can be expressed in the following compact form:

F a =π

4l2ρ[(Gn − Vn

)n+Gτ τ

],

Ma(H) =π

128l4ρ (α−Ma) .

(28)

IV. Aeroelastic Model

A scheme of the typical section is depicted in Fig. 2. The mass and elastic centers are assumed at themidchord point, but the model can be easily generalized. The section moves under the aerodynamic loadsand elastic reactions, the latter caused by the bending and torsional springs kx = mω2

x, kz = mω2z , and

kα = Jα ω2α. The quantities m and Jα are the mass and moment of inertia (about the elastic center) per

unit span length, whereas ωx, ωz and ωα denote the uncoupled natural bending and torsional frequencies.

Figure 2. Typical section aeroelastic model.

Introduced the ratios between the virtual mass and moment of inertia and those of the airfoil

σ :=π4 l

2 ρ

m, µ :=

π128 l

4 ρ

Jα, (29)

the equations of motion are written asg

H = σ[(Gn − Vn

)n+Gτ τ

]+F e

m,

α = µ (Ma − α)− Me

Jα,

(30)

gIt should be noted that the angle of attack is clockwise positive, whereas the acting moments are counter-clockwise positive.

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where, denoted by He = Hxe + i Hze and αe the point and angle of vanishing elastic reactions, the forceand moment due to the springs are F e = −mω2

x (Hx −Hxe)− i mω2z (Hz −Hze) and Me = Jα ω

2α (α− αe).

To obtain an explicit formulation for the equations of motion, the quantity Vn is deduced from Eq. (30) inthe following way (n = −α τ , F e = F e

n n+ F eτ τ ):

Vn = H · n− H · τ α

=1

1 + σ

(σGn +

F en

m− αVτ

). (31)

Substituting Eq. (31) into Eq. (30), the equations of motion finally achieve the form

H =1

1 + σ

[σ (Gn + αVτ ) +

F en

m

]n+

(σGτ +

F eτ

m

)τ ,

α =1

1 + µ

(µMa − Me

Jα

),

(32)

with initial conditions H(0) = H0, α(0) = α0, H(0) = H0, α(0) = α0, and Γb(0) = Γb0 .

V. Numerical Results

The present model is implemented in FORTRAN 77. The equations of motion and vortex kinematics areintegrated by means of a fourth-order Runge-Kutta scheme, whereas the aerodynamic loads are analyticallyevaluated at each sub-step using Eq. (28). Once a preliminary validation is carried out, the computationalcode is used to simulate the arbitrary motion of the typical section and its aeroelastic behavior.

A. Wagner Problem

Model validation is achieved by simulating the lift response due to a sudden start from rest. The numericalresults are compared with the theoretical prediction by Wagner,3 expressed in terms of the so-called Wagnerfunction.3 The latter models the wake effect, which causes a lag in the lift generation.

Figure 3. Normalized lift time-history due toa sudden start from rest.

Since Wagner3 tackled the problem of the sudden startof a thin airfoil by assuming small angles and flat wake, thesame approximations are inserted into the present numericalcode. The lift is evaluated as the linearized normal compo-nent of the aerodynamic force. Its time-history is normalizedby the linearized steady-state value (ρπlu2

∞α) to be directlycompared with Wagner function.3

The simulation data are α = 1, u∞ = 20m/s, β = 0,and l = 1m. The normalized lift time-history is comparedwith Wagner function in Fig. 3. The numerical results arefound to match the theoretical prediction, apart from a slightdifference at the beginning of the simulation. This is due tothe fact that Wagner function is equal to 0.5 for t = 0+,namely the lift suddenly takes half the steady-state valueaccording to the theory. Since wake effects are taken intoaccount by vortex shedding, the present model cannot givesuch a discontinuity. Indeed, the simulation gives zero lift at the initial time, because no point vorticeshave been already released from the trailing edge. However, Fig. 3 shows that the numerical results are inexcellent agreement with the theoretical prediction for t ≥ 0.1 s.

B. Aeroelastostatic Response

In order to numerically simulate the aeroelastic response of the system and the related transient phase, thetypical section is initially assumed in elastic equilibrium (H0 = He, α0 = αe, Γb0 = 0). Then, the incomingflow moves the section to a stationary configuration in which the aerodynamic loads balance the elasticreactions. Firstly, a few analyses are carried out by assuming that the section is only able to translate,

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namely α ≡ α0. Then, the response of the three degree-of-freedom system is explored, with the aim to pointout the effects of aeroelastic coupling between bending and torsion. The assumed section properties are

σ = 0.1, µ = 0.05, ωx = 2π · 12.5 rad/s, ωz = 2π · 2.5 rad/s, ωα = 2π · 5 rad/s, Hxe = Hze = 0 .

The time-histories of the vertical position of the midchord point in the case of no pitch are presentedin Fig. 4. The parametric studies show the influence of u∞ and α0 on the transient and aeroelastostaticresponses. It can be appreciated that a positive aerodynamic damping arises during the transient phase.This is due to the fact that the normal component of the aerodynamic force is linear in the body circulation,which, neglecting wake effects and assuming no pitch, is estimated as Γb ≈ −πl [u∞ sin (α+ β)− Vn]h.Hence, the body circulation decreases in modulus when the section moves up, whereas it increases when thesection moves down. The consequent variations in the aerodynamic force act as a damping on the system.This effect is enhanced for increasing u∞, whereas the oscillation around the steady-state value is found togrow with α0. Since the stationary aerodynamic force is given by ρπlu2

∞ sin (2α0) /2, the aeroelastostaticresponse is consequently affected by both α0 and u∞.

(a) α0 = 5, u∞ = 10, 20, 30m/s (b) u∞ = 10m/s, α0 = 5, 10

Figure 4. Aeroelastostatic response (no pitch): vertical position of the midchord point.

(a) Hz(t) (b) α(t)

Figure 5. Aeroelastostatic response (with pitch): vertical position of the midchord point and angle of attackfor α0 = 5, u∞ = 10, 15m/s.

Pitch completely changes the aeroelastic behavior of the section. As can be ascertained from Fig. 5, themost relevant effect is that the aerodynamic damping now decreases with u∞. Thus, the typical sectionis likely to experience flutter, because of the aeroelastic coupling between bending and torsion. This phe-nomenon causes the time-histories to show multiple harmonic contributions. Pitch also affects the systemstiffness, namely the aeroelastostatic response. Since the stationary angle of attack is now larger than theinitial one, the vertical displacement of the midchord point increases with respect to the case of no pitchwith all the other parameters being the same (see Figs. 4 and 5).

hThe condition of vanishing tangential aerodynamic force F aτ = 0 is the square of a binomial in the absence of wake.According to Eqs. (25) and (26), a first estimate of the body circulation is thus Γb ≈ −πl [u∞ sin (α+ β)− Vn + αl/4].

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(a) No pitch

(b) With pitch

Figure 6. Comparison between the wake configurations at the timet = 2 s for α0 = 10, u∞ = 10m/s.

Finally, observe the wake patternsdepicted in Fig. 6. Assumed thesame simulation data, the configu-ration above is obtained by takingα ≡ α0, whereas the other one alsoincluding pitch. A strong startingvortex is observed in both cases, dueto the fact that the initial circulationaround the section is zero. However,the wake evolution is different duringthe transient phase, because of pitcheffects. It can be concluded that theflat-wake assumption is not reason-able if significant pitch occurs.

C. Aeroelastic Response to an Impulse

The last simulations explore the aeroelastic response to an impulsive load. The typical section is assumedin the steady-state configuration previously obtained for u∞ = 10m/s, α0 = 5, namely Hxss

≈ 0, Hzss ≈0.017m, αss ≈ 5.96 (see Fig. 5). The impulse is applied as an initial horizontal or vertical velocity, with theaim to point out the different role played by the horizontal and vertical translations in vorticity shedding.

(a) Hx0 = 2m/s (b) Hz0 = 2m/s

Figure 7. Aeroelastic response to an impulse: vertical displacement of the midchord point.

Figure 7 presents the time-histories of the vertical displacement of the midchord point with respect to theequilibrium position. Horizontal impulses excite the lag degree-of-freedom (Hx). This causes high-frequencyvariations in the aerodynamic force, responsible for the corresponding harmonics in the time-histories of thevertical displacement (left-hand side of Fig. 7). On the contrary, vertical impulses do not involve appreciablehorizontal motion. Such a result can be explained by observing that the aerodynamic force is substantiallyvertical for moderate angles of attack. Moreover, the in-plane bending frequency is larger than the out-of-plane one, as actually occurs for wing sections. Consequently, relevant lag oscillations only result from initialconditions or external loads with a significant x-component. Indeed, the horizontal motion has been foundto be negligible in the preceding simulations. For this reason, it has not been mentioned in the discussion.Because of the small x-component of the aerodynamic force, horizontal oscillations are basically undampedby aerodynamics. In the absence of structural damping, they thus require a large amount of time to vanish.

Figure 8. Wake configuration at the time t = 3 s for Hx0 = 2m/s.

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Figure 9. Wake configuration at thetime t = 0.8 s for Hz0 = 2m/s.

Vortex shedding depends on the tangential velocities at thetrailing-edge. At moderate angles of attack, these are stronglyinfluenced by horizontal motion. Indeed, the wake pattern is or-ganized in dipoles in the case of an horizontal impulse (Fig. 8),because of the high-frequency variations in the shed vorticity. Onthe other hand, a vertical impulse implies a sudden change in thebody circulation, which results in a strong starting vortex (Fig. 9).Since vertical motion is highly damped by aerodynamics, dipolestructure appear progressively later in this latter case.

VI. Concluding Remarks

The aim of the present work was to derive a low-order free-wake aeroelastic model suitable for preliminarydesign, sensitivity analysis and optimization of aircraft configurations undergoing large-amplitude deflections.This has been achieved by coupling a fully nonlinear unsteady aerodynamic model based on conformalmappings with the equations of motion for a typical section.

After a preliminary validation, numerical simulations have demonstrated the capability of the proposedmodel to capture arbitrary motions and consequent wake patterns. The analytical evaluation of the aerody-namic loads has simplified the physical interpretation of the results and reduced the computational burdenin comparison with aeroelastic models involving numerical integration of pressure.

Further improvements will include chord-wise flexibility, which requires the evaluation of generalizedaerodynamic forces associated with elastic degrees-of-freedom. Possible applications of the present modelinvolve aeroelastic design of highly flexible wings, investigation on the onset of limit-cycle oscillations, quan-tification of the energy exchange between fluid and structure and its parametric dependency on the sectionproperties, flapping-wing propulsion and blade-vortex interaction problems.

Acknowledgments

I would like to thank my advisors Prof. Franco Mastroddi and Prof. Giorgio Riccardi, who followed thedevelopment of my Master’s thesis and constantly supported me with their experience.

References

1Bisplinghoff, R. L., Ashley, H., and Halfman, R. L., Aeroelasticity, Dover Publications, Mineola, NY, 1996.2Morino, L., Mastroddi, F., “Introduction to Theoretical Aeroelasticity for the Aircraft”, Course Notes on Aeroelasticity,

Master Course in Aeronautical Engineering, “Sapienza” Universita di Roma, Rome, Italy, 2014.3Wagner, H., “Uber die Entstehung des Dynamischen Auftriebes von Tragflugeln”, ZAMM, Vol. 5, No. 1, 1925, pp. 17-35.4Kussner, H. G., “Zusammenfassender Bericht uber den instationaren Auftrieb von Flugeln ”, Luftfahrtforschung, Vol. 13,

No. 12, 1936, pp. 410-424.5Theodorsen, T., “General Theory of Aerodynamic Instability and the Mechanism of Flutter”, NACA Rep. 496, 1935.6Theodorsen, T., Garrick, I. E., “Mechanism of Flutter. A Theoretical and Numerical Investigation of the Flutter Problem”,

NACA Rep. 685, 1940.7Davis, P. J., The Schwarz Function and its Applications, Carus Mathematical Monographs, Mathematical Association

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10Ablowitz, M. J., and Fokas, A. S., Complex Variables: Introduction and Applications, 2nd ed., Cambridge UniversityPress, Cambridge, England, UK, 2003.

11Sarpkaya, T., “An Inviscid Model of Two-Dimensional Vortex Shedding for Transient and Asymptotically Steady Sepa-rated Flow Over an Inclined Flat Plate”, Journal of Fluid Mechanics, Vol. 68, No. 1, 1975, pp. 109-128.

12Kiya, M., and Arie, M., “A Contribution to an Inviscid Vortex-Shedding Model for an Inclined Flat Plate in UniformFlow”, Journal of Fluid Mechanics, Vol. 82, No. 2, 1977, pp. 223-240.

13Glegg, S. A. L., Devenport, W., “Unsteady Loading on an Airfoil of Arbitrary Thickness”, Journal of Sound andVibration, Vol. 319, Nos. 3-5, 2009, pp. 1252-1270.

14Riso, C. “Free-Wake Modelling via Conformal Mappings for the Aeroelastic Analysis of a Typical Section”, M.Sc. Thesis,Dept. of Mechanical and Aerospace Engineering, “Sapienza” Universita di Roma, Rome, Italy, 2014.

15Minotti, F. O., “Unsteady two-dimensional theory of a flapping wing”, Physical Review E - Statistical, Nonlinear, andSoft Matter Physics, Vol. 66, No. 5, 2002, pp. 1-10.

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