Acoustic emission of aeroelastic vortex-gust interactions
Huansheng Chen(1) , Justin W. Jaworski(2)(1) Lehigh University, USA, huc216@lehigh.edu(2) Lehigh University, USA, jaworski@lehigh.edu

Abstract
The coupled interaction between an unsteady vortical ﬂow and dynamics of an aerodynamic structure is a canonical problem for which analytical studies have been typically restricted to either static or prescribed structural
motions. The present effort extends beyond these restrictions to include a Joukowski airfoil on elastic supports
and its aeroelastic inﬂuence on the incident vortex, where it is assumed that all vorticity in the ﬂow ﬁeld can be
represented by a collection of line vortices. An analytical model for the vortex motion and the unsteady ﬂuid
forces on the airfoil is derived from inviscid potential ﬂow, and the evolution of the unsteady airfoil wake isgoverned by the Brown and Michael equation. The aerodynamic sound generated by the aeroelastic interactionof an incident vortex, shed Brown-Michael vortices, and the moving airfoil is estimated for low-Mach-number
ﬂows using the Powell-Howe acoustic analogy.
Keywords: Vortex sound, conformal mapping, Brown and Michael equation

1

INTRODUCTION

The encounter of a vortex gust with an aerodynamic body is a canonical ﬂuid-structure interaction with implications for the prediction of transient loads on ﬂiers and swimmers and their generation of vortex sound. For
theoretical analyses in particular, the representation of how vorticity is shed into the wake to satisfy the Kutta
condition at the trailing edge plays a crucial role in gust-airfoil interactions and related unsteady airfoil problems. An early model to account for the effect of vortex shedding of a delta wing was formulated by Brown
and Michael (1). Their model supposes that the vorticity shed from an edge rolls up via a connecting vortex
sheet into a point vortex with time-varying circulation. We refer to such vortices as Brown-Michael vortices,
whose circulation is set instantaneously to satisfy the Kutta condition. The Brown and Michael equation describes the motion of the vortex tethered to the trailing edge before it is released as a free vortex into the ﬂow.
However, the original formulation does not guarantee the removal of spurious unbalanced couple on the airfoil,
which is of minor concern to the ﬂuid dynamic problem but is important for aeroacoustic predictions based on
the model. Howe (2) later constructed an emended version of the Brown and Michael equation that eliminates
this unbalanced couple.
Previous analytical efforts to analyze gust-airfoil interactions are typically restricted to either static or prescribedairfoil motion, with the exception of a few recent works (3, 4, 5). The present work extends to study the
dynamical problem and the aerodynamic sound problem of a gust vortex past by a Joukowski airfoil underaeroelastic motions. The dynamical problem is formulated analytically and evaluated numerically using a timedependent conformal mapping. The Brown and Michael framework models the unsteady shedding of vorticity
from the airfoil into the wake, and the linear aeroelastic plunging motion of the airfoil is calculated using
unsteady airfoil theory.
The remainder of the paper is organized as follows. Section 2 outlines the model scenario and its dynamic formulation studied previously by Chen and Jaworski (6) and presents the vortex sound model. Section 3 validates
the analytical/numerical framework against previous acoustic results for a stationary ﬂat plate by Manela (4).
New results are then presented for the acoustic emission of a (non-ﬂat) Joukowsky airfoil that is free to moveaeroelastically under the inﬂuence of the vortical ﬁeld. Section 4 contains concluding remarks for this study.

604




∑













Figure 1. Schematic of the generalized model problem of an incident vortex interaction with a symmetric
Joukowski airfoil on elastic translational support in a uniform ﬂow, where U is the ﬂow speed and h(t) denotes
the displacement of the airfoil. The airfoil can be considered as a damped harmonic oscillator with mass m,
damping coefﬁcient cd , and spring stiffness kh . Γ denotes the incident line vortex, and γn is the tethered
trailing-edge vortex whose motion is determined by the emended Brown and Michael equation. Σ is the set offree vortices generated at the airfoil trailing edge due to unsteady airfoil loads in response to the incident line
vortex or the airfoil motion. The airfoil has zero angle of attack.

2

MATHEMATICAL FORMULATION

Figure 1 illustrates the model problem of a Joukowski airfoil on elastic supports in two-dimensional uniform
ﬂow with an incident line vortex Γ and the vorticity ﬁeld Σ shed into the wake. All vortices have positive
circulation in the anticlockwise direction. The strength of the trailing-edge vortex γn satisﬁes the Kutta condition, and its motion obeys the emended Brown and Michael equation (2). The mathematical formulation of theaeroelastic system is now described.
2.1 Mapping
The conformal mapping of the Joukowski airfoil between the physical z-plane and the mapped ζ -plane is described byζ (z) =

1  2z + z − 4λ 2 − f0 .
2

(1)

Using Eq. (1), the Joukowski airfoil in the physical z-plane (z = x + iy) with its trailing edge locating at (2λ , 0)is mapped to a circle with radius r = 1 in the f -plane ( f = f1 + i f2 ), as shown in Fig. 2. Note the offset ofthe circle center at f0 = fx0 + i fy0 and the corresponding trailing edge at (λ , 0). For the symmetric Joukowski
airfoils considered in this work, fy0 = 0. The unit circle in the f -plane is then shifted by an elementary mappingto the origin in the ζ -plane.
When time-dependent airfoil motions are considered, Eq. (1) becomesζ (s) =


1s(z,t) + s2 − 4λ 2 − f0 ,2(2)

where s(z,t) = z − ih(t). The airfoil displacement h(t) may be either prescribed or be part of the solution, as in
the case of aeroelastic airfoil motions.

605ζ   






ζ








   


 

 

  



Figure 2. Successive mappings of a generalized Joukowski airfoil in the physical z-plane to a unit circle centeredat the origin in the ζ -plane, T is the trailing edge location. The trailing-edge position through the successive
mappings is indicated by the red dot.
2.2 Flow complex potential
The complex potential of the ﬂow isλ2w(ζ ) = wγ + wΓ + wh +U ζ + f0 +,
ζ + f0(3)

where wγ (ζ ), wΓ (ζ ), and wh (ζ ) are the contributions due to the shed and free vorticity ﬁeld, the incident
vortex, and the airfoil motion, respectively. The last term of Eq. (3) represents the uniform background ﬂow of
speed U.
The complex potential of the incident vortex wΓ (ζ ) is determined from the model of a vortex moving around acylinder (7), which yields


1
iΓ
iΓ
iΓ
wΓ (ζ ) = − log (ζ − ζΓ ) +log ζ − ∗ −logζ ,
(4)
2π
2π
ζΓ
2π
where the last two terms guarantee that the total circulation inside the cylinder is initially zero, and the superscript ∗ denotes the complex conjugate. Similarly, the complex potential of the vortices shed into the wake
is

n
1
iγk
iγk
log ζ − ∗wγ (ζ ) = ∑ − log ζ − ζγk +
.
(5)
2π
2π
ζγk
k=1
By appeal to Kelvin’s theorem, the bound vorticity inside the cylinder at any time has total circulation − ∑nk=1 γk ,which is used to compute the aerodynamic lift force.
The complex potential of the airfoil motion is (8),1wh (ζ ) = iV ζ −,(6)ζwhere V = dh/dt is the instantaneous velocity of the airfoil in the downward direction.
2.3 Vortex shedding and the emended Brown and Michael equation
The motions of the vortices shed most recently by the trailing edge are described by the emended Brown andMichael equation (2),dxγn
Ψj dγn· ∇Ψj +
= vγn · ∇Ψj ,
dt
γn dt

606

j = 1, 2,(7)

where xγn represents the location of a shed vortex tethered to the trailing edge with circulation γn , and vγn is
the ﬂuid velocity when the local velocity induced by γn is excluded. Here a Cartesian coordinate system isconsidered, where x ≡ (x, y). Ψj (x,t) denotes the stream function of the complex potential of the ﬂow in the
j-direction. For a Joukowski airfoil mapped to the ζ -plane, the components of the stream function are (9)Ψ1 = Im {ζ + 1/ζ }and Ψ2 = Im {−i (ζ − 1/ζ )} .

(8)

The instantaneous circulation of the tethered vortex γn (t) is obtained by enforcing the Kutta condition at thetrailing edge of each instant in time (6),

 2
 ∗
ζγ  − 1T ζγ − 12 2Γ (1 − Re{T ∗ ζΓ }) n−1n∗γn (t) =  2(9)
− ∑ γk  k
2 − 2πV Re{T } ,2∗ζγ  − 1|T ∗ ζΓ − 1|T ζγ − 1k=1nkin which T ∗is complex conjugate of the trailing edge T location (T = λ − f0 ) in the ζ -plane. The tethered
trailing-edge vortex is released and becomes a free vortex when dγn /dt changes sign, at which time anothertethered vortex is placed at the airfoil trailing edge whose motion and instantaneous circulation are determinedby Eq. (7) and Eq. (9), respectively.
Equation (7) can be rearranged into the equivalent scalar form
dz∗γn
1 dγn
+ (H1 − iH2 )= v∗γn ,
dt
γn dt

(10)

where z∗γn = x − iy and v∗γn = vx − ivy . The functions H1 and H2 involve the stream functions Ψj and theirderivatives. Equation (10) is the equivalent scalar form of the emended Brown and Michael equation, which is
employed for the theoretical analysis in this work. Speciﬁc details related to the derivation of Eq. (10) and theexpressions of H1 and H2 from (6) are presented in the Appendix for reference. Also, the complex velocity ofthe shed vortex with self-potential velocity excluded is (7)v∗γn = −iγn ζ  (zγn )+ Fγn (zγn ),4πζ  (zγn )where
Fγn (zγn ) = ζ dw iγn 1+.
dζ 2π ζ − ζγn

(11)

(12)

2.4 Kinematics of the incident and free vortices
Similarly, the complex velocity of the incident line vortex at sΓ is (7)ds∗ΓiΓζ  (sΓ )=−+ F  (sΓ ),dt
4πζ  (sΓ )whereFΓ (sΓ ) = ζ dw iΓ 1+.
dζ 2π ζ − ζΓ

(13)

(14)

Also, the equation of motion for each of the n − 1 free vortices isds∗γkiγk ζ  (sγk )=−+ Fγk (sγk ),dt
4πζ  (sγk )where
Fγk (sγk ) = ζ dw iγk1+.
dζ 2π ζ − ζγk

(15)

(16)

607

2.5 Airfoil motion and loads
The airfoil moves aeroelastically under its lift force and the equation of motion of the elastic mount,
m

d2h
dh
+ kh h = −L ,
+ cd
2
dt
dt

(17)

where h(t) denotes the vertical displacement of the airfoil (postive downward). The airfoil suspension is modeled as a linear harmonic oscillator with mass m, damping coefﬁcient cd , and spring stiffness kh . The unsteadylift force L is determined by (10):L = ρUΓa (t) + ρ

 c
0

∂
Γa (x,t) dx .
∂t

(18)

The system of dynamical equations is formed from Eqs. (9), (10), (13) and (15), which consists of 2(n + 2)
ﬁrst-order ordinary differential equations for the position (sxΓ (t), syΓ (t)) of the incident line vortex, the positions(sxγn (t), syγn (t)) of n trailing-edge vortices, and the instantaneous airfoil displacement h(t) and the velocity V (t).
The system of equations is marched forward in time using a fourth-order Runge-Kutta algorithm. Once solved,
the results are mapped into the physical z-plane by using z(t) = s(t) + ih(t).
2.6 Acoustic emission
The aerodynamic sound problem at low Mach numbers can be solved by the Powell-Howe acoustic analogy (7,
11)


1 ∂2
dV
2
δ (y),(19)−∇
B = div(ω ∧ v) +dt
c20 ∂t 2
where B is the stagnation enthalpy per unit mass of the ﬂuid (12) and c0 is the isentropic speed of sound. The
ﬁrst term on the right hand side denotes the vorticity contribution via the divergence of the Lamb vector, andthe second term is the airfoil motion contribution. In the acoustic far ﬁeld, the acoustic pressure and stagnation
enthalpy are related by the linearized approximationp(x,t) ≈ ρB(x,t).

(20)p(x,t) = pv (x,t) + pω (x,t),(21)

The acoustic pressure can be written as

where pv (x,t) is the acoustic pressure due to the airfoil motion,2ρλ 2 cos θ ∂ 2
pv (x,t) ≈ 
2c0 |x| ∂t 2

 [t]dh / dtdτ ,−∞[t] − τ(22)and pω (x,t) denotes the contribution from all vortices including the incident vortex and shedding vortices, which
can be calculated using Howe’s formula (7), −ρωx j∂ [t] dx ∂Y j dy ∂Y j
dτ

−
.
(23)
pω (x,t) ≈ √(3/2) ∂t −∞ dτ ∂ ydτ∂x[t] − τ2π 2c0 |x|z(τ)
Here ω is the vortex strength, x j /|x| denotes the observer direction θ , where sin θ = x1 /|x| and cos θ = x2 /|x|,x is the observer location in the far ﬁeld (|x| → ∞), [t] = t − |x|/c0 is the acoustic retarded time, t denotes
the observer time, and τ is the source time. Y j ( j = 1, 2) are the components of the Kirchhoff vector that are
established from the conformal mapping,Y1 = Re (ζ + 1/ζ )and Y2 = Re {−i (ζ − 1/ζ )} .

(24)

The integrals in Eq. (23) can be evaluated numerically once the path of the vortex z(τ) = x(τ) + iy(τ) has been
determined.

608

3

RESULTSIn this section we ﬁrst compare the results of a simpler case against Manela’s work (3) to validate the mathematical framework, then plot new ﬁndings for the aeroelastic case.
 The model√ equations are nondimensionalized
by x = x/(2λ ), y = y/(2λ ), t = Ut/(2λ ), and Π([t]) = p(x,t) 8|x|/(ρU 2 M), which produce the following
parametric groups that deﬁne the numerical simulations (13):Γ=4ρa ltcdΓ2λ ωn, ωn =
, μ=
, ξ=
.
4πUλ
U
πρc
2mωn


The last three parameters are related to the airfoil suspension, where ρa is the airfoil density, ωn = kh /m is thenatural frequency of the suspension. The airfoil chord length and λ are related by c = 3λ −2 fx0 −λ 2 /(2 fx0 −λ ),and M = U/c0 is the Mach number.
3.1 Veriﬁcation of simulations
Consider the degenerate case of the generalized model in Fig. 1 of a ﬁxed, ﬂat airfoil of length 4λ immersed
in a uniform ﬂow with speed U in the x-direction. An incident line vortex of constant strength Γ = 0.2 isreleased into the ﬂow at the initial instant in time (t = 0) at position (x0 , y0 ) = (−20, 0.2) and advects freely
over the airfoil. This scenario has been previously investigated by Manela (4) and furnishes a veriﬁcation casefor the more general framework developed in this paper. Figure 3(a) illustrates the comparison of the numericalresults for the nondimensional acoustic pressure measured directly above a stationary ﬂat plate or a stationary
Joukowski airfoil at θ = π/2, where the solid curve agrees with Fig. 3(c) of Manela (4).
3.2 Aeroelastic case
Now consider the case of an incident vortex past by a Joukowski airfoil under aeroelastic motion, as illustrated
in Figure 1. The Joukowski airfoil has 12% thickness and zero angle of attack. The initial strength andlocation of the incident vortex are the same as in Section 3.1, and the additional dimensionless parameters forthe aeroelastic case are the damping ratio ξ = 0, reduced natural frequency ω n = 0.5, and mass ratio μ = 10.
The sound produced by the unsteady suction (θ = π/2) in different aspects is shown in Figure 3(b) and (c),respectively. In Figure 3(b), the solid curve denotes the incident vortex acoustic contribution, and the dashedcurve denotes the wake contribution which includes the effect of 12 Brown-Michael vortices that are shed by
the trailing edge over the course of the simulation. The reduced pressure of the incident vortex contribution
decreases when the incident vortex passes by the leading edge, and the pressure of the wake contribution starts
to ﬂuctuate due to more vortices shedding from the airfoil trailing edge. After the incident vortex passes by
the trailing edge, the wake contribution and the incident vortex contribution start to balance together with same
strength in opposite signs and yield a net zero total acoustic pressure in long times, as illustrated in Figure3(c). Compared with the total sound for a stationary ﬂat and thick Joukowski airfoils in Figure 3(a), the resultin Figure 3(c) shows that aeroelastic effects have a modest slight inﬂuence in the total sound for the parametersconsidered here. However, the comparison of ﬂat airfoil and Joukowski airfoil in Figure 3(a) shows that the
airfoil thickness controls signiﬁcantly the peak noise due to the acceleration of the vortex in the leading-edge
interaction. Figure 3(d) presents the sound observed along a different observation angle (θ = 0) produced by
the unsteady lift contributions of the incident vortex, the wake, and the airfoil motion.

4

CONCLUSIONSAn analytical model for the aeroelastic interaction of an incident vortex and a symmetric Joukowski airfoil isformulated in this paper. The aeroelastic vortex dynamics study by (6) is extended to study the corresponding
acoustic emissions using the Powell-Howe acoustic analogy. The acoustic signature for the special case of astationary ﬂat-plate airfoil is veriﬁed against available results in the literature. New acoustic results show thatthe airfoil thickness controls signiﬁcantly the total sound, while a modest difference in total sound is observed609

Leading
Edge

Trailing
Edge
TrailingEdge(d)Leading
Edge(c)
TrailingEdge(b)

LeadingEdge(a)Figure 3. Acoustic radiation of a line vortex encountering a stationary ﬂat-plate airfoil or a Joukowski airfoil
with 12% thickness: (a) comparison of total acoustic radiation for the stationary ﬂat and Joukowski airfoils
along θ = π/2; (b) acoustic contributions of incident vortex (ΠΓ ) and wake (Πγ = ∑k=12
k=1 Πγk ) for a Joukowski
airfoil under aeroelastic motion along θ = π/2; (c) total normalized acoustic pressure for a Joukowski airfoil
under aeroelastic motions observed at angle θ = π/2; (d) acoustic contributions of the incident vortex (ΠΓ ),vortical wake (Πγ ), and the airfoil motion (Πv ) for a Joukowski airfoil under aeroelastic motion, as observed
along θ = 0. For the aeroelastic simulations in (b-d), ξ = 0, ω n = 0.5, and μ = 10.

610

in the comparison of stationary and aeroelastic cases for the parameters considered. A complete parameter
sweep of the aeroelastic cases is the subject of ongoing research to determine conditions for minimum noiseproduction and the associated dynamics of the gust-airfoil interaction.

ACKNOWLEDGEMENTS
This work was supported in part by the Air Force Ofﬁce of Scientiﬁc Research under AFOSR grant FA955015-1-0148, monitored by Drs. Douglas Smith and Gregg Abate, and by the National Science Foundation underawards 1805692 and 1846852, monitored by Dr. Ronald Joslin.

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