Theoretical solutions for unsteady compressible … · Theoretical solutions for unsteady compressible subsonic ... (with applications in aeroelasticity) ... Fung [41], Bisplinghoff

MATHEMATICS IN ENGINEERING, SCIENCE AND AEROSPACEMESA - www.journalmesa.com

Vol. 2, No. 1, pp. 1-27, 2011c© CSP - Cambridge, UK; I&S - Florida, USA, 2011

Theoretical solutions for unsteady compressible subsonicflows past oscillating rigid and flexible airfoils

Dan Mateescu

Doctor Honoris Causa, FCASI, AFAIAA, Erskine FellowProfessor, Aerospace Program, Mechanical Engineering DepartmentMcGill University, Montreal, QC, H3A 2K6, CanadaE-mail: [email protected]

Abstract. This paper presents simple and efficient analytical solutions for unsteady compressiblesubsonic flows past flexible airfoils executing low frequency oscillations. These analytical solutionsare obtained with a method using velocity singularities related to the airfoil leading edge and ridges(defined by the changes in the boundary conditions). Efficient analytical solutions in closed form arepresented for the unsteady lift and pitching moment coefficients and for the chordwise distribution ofthe unsteady pressure difference coefficient in the general case of rigid or flexible airfoils executingoscillatory rotations and normal-to-chord translations and flexural oscillations. The method has beenvalidated for the pitching and plunging oscillations of the rigid airfoils by comparison with resultsbased on Jordan’s data for compressible flows and by comparison with the solutions obtained byTheodorsen, Postel & Leppert and Mateescu & Abdo for the limit case of incompressible flows. Adetailed analysis of the variations of the aerodynamic coefficients with the Mach number and withthe reduced frequency of oscillations is also presented for the cases of oscillatory pitching rotationsand normal-to-chord translations of rigid airfoils and for the flexural oscillations of flexible airfoils.Efficient analytical solutions are presented for the flexural oscillations of airfoils in compressibleflows, which can be efficiently used in solving aeroelastic interaction problems.

1 Introduction

The steady and unsteady flows past airfoils (or wing sections) have been extensively studied in the lastseven decades for their aeronautical applications. For incompressible flows, the method of conformaltransformation has been initially used to obtain steady flow solutions for particular airfoil shapes,such as Joukowski, Karman-Trefts, Betz-Keune, von Mises and Carafoli airfoils [1-5]. The classicalthin airfoil theory developed by Glauert and Birnbaum [1-6] established the foundations of the linearaerodynamics of thin airfoils of arbitrary shapes in incompressible flows, by using a modified Fourier

2010 Mathematics Subject Classification: 76G25, 76M40, 76N99, 31A99Keywords: Unsteady Aerodynamics, Subsonic compressible flows, Oscillating airfoils, Shedding vortices.

2 Dan Mateescu

series expansion for the distributed vortex intensity along the chord and a truncated Fourier represen-tation of the airfoil camberline. However, the Fourier series are not convenient to model the airfoilswith discontinuities in the boundary conditions, such as the flapped airfoils. In order to overcomesome of the deficiencies of the methods based on Fourier series expansions, several authors, such asWeber [7-8] have developed linear methods using velocity singularities, and others, such as Munk[9] and Cheng & Rott [10] mathematically derived inversion formulae for the linear analysis of thinairfoils in incompressible flows based on small perturbation assumption.

An efficient method using special singularities in the expression of the fluid velocity related to sin-gular points on the airfoil (the leading edge and ridges - points where the boundary conditions displaychanges) and leading to simple analytical solutions in closed form has been developed by Mateescu[11], Mateescu & Newman [12] and Mateescu & Nadeau [13] for airfoils in steady incompressibleflows and by Mateescu & Abdo [14] for unsteady flows past oscillating rigid and flexible airfoils. Thismethod has proven to be very efficient also in solving problems of unspecified geometry involvingadaptive surfaces, such as flexible-membrane and jet-flapped airfoils [12]. A similar method based onvelocity singularities has also been developed by Mateescu [15] for wing-body systems in supersonicflow.

An efficient second-order analytical method of solution, taking into account the real physicalvelocity behavior at the leading and trailing edges (such as finite velocity at the leading edge), hasbeen developed by Mateescu & Abdo [16] for airfoils in incompressible and compressible subsonicflows. The analytical solutions thus obtained, which included also the viscous effects on the pressuredistributions on the airfoils, were found in good agreement with the numerical and experimentalresults for various Reynolds and Mach numbers.

Other solutions involving intensive numerical calculations have been obtained for steady flowsusing conformal mappings (Sells [17], Halsey [18], Ives [19], Bauer et al. [20] ), or using boundaryelement methods based on source, doublet and vortex panels (see Kuethe & Chow [4], Hess & Smith[21], Hunt [22], Katz & Plotkin [23] and Mateescu [11] ).

More recently, computational solutions have been obtained using various numerical methods forsolving the Euler or Navier-Stokes equations, such as those based on finite-difference or finite-volumeformulations (for examples see Anderson [24], Cebeci [25, 26], Drela & Giles [27, 28], Jameson[29], Elrefaee et al. [30], Nelson et al. [31] and Mateescu & Stanescu [32] ). Several authors solvednumerically the viscous flows past airfoils or wings by combining the inviscid Euler solvers, or panelmethods, with the analysis of the boundary layer developed along the airfoil contour (see Cebeci [25,26] and Drela & Giles [27, 28] ). Recently, a complete analysis of the viscous flows past airfoils at lowReynolds numbers based on the accurate time-integration of the Navier-Stokes equations (includingthe analysis of the trailing edge flow separation) has been performed by Mateescu and Abdo [33].

The analysis of the unsteady flows past oscillating airfoils and wings has been mostly motivatedby the efforts made to avoid or reduce undesirable unsteady effects in aeronautics, such as flutter,buffeting, and dynamic stall. Potentially beneficial effects of these unsteady flows have also beenstudied, such as propulsive efficiency of flapping motion, controlled periodic vortex generation, stalldelay, and optimal control of unsteady forces to improve the performance of turbomachinery, heli-copter rotors, and wind turbines. The foundations of the unsteady aerodynamics of oscillating airfoilshave been established by Theodorsen [34], Theodorsen & Garrick [35], Wagner [36], Kussner [37,38], and von Karman & Sears [39], who studied the unsteady flow past an oscillating thin flat plateand a trailing flat wake of vortices in incompressible flows (see also Carafoli [2] ). Further studiesinvolving detailed unsteady flow solutions of oscillating airfoils (with applications in aeroelasticity)have been performed by Postel and Leppert [40], Fung [41], Bisplinghoff & Ashley [42], McCroskey

Theoretical solutions for unsteady compressible subsonic flows past oscillating rigid and flexible airfoils 3

[43,44], Dowell et al. [45, 46], Kemp & Homicz [47], Basu & Hancock [48], and others.[49 - 53].Some of the recent studies used panel methods or numerical methods based on finite difference orfinite volume for these unsteady aerodynamic problems (see Dowell [45], Mateescu et al. [54].

The aim of this paper is to present simple and efficient analytical solutions for unsteady com-pressible subsonic flows past flexible airfoils executing low frequency oscillations. These analyticalsolutions are obtained with a method using velocity singularities related to the airfoil leading edgeand ridges (defined by the changes in the boundary conditions on the airfoil). This method of veloc-ity singularities (different from Theodorsen’s method for incompressible flows based on singularitiesin the velocity potential) represents an extension to compressible flows of the method developed forincompressible flows by Mateescu & Abdo [14].

The analytical solutions obtained with this method are simple, efficient and in closed form (incontrast with complicated previous solutions based on Fourier series expansions, approximate poly-nomial fitting for each Mach number and on the numerical evaluations of several integrals (such as in[51, 52] ). These efficient solutions are particularly suitable for the aeroelastic studies, in which theunsteady aerodynamic analysis is performed in conjunction with the analysis of the related structuralmotion involving oscillatory flexural deformations. Although potentially more accurate, a completenumerical approach to solve simultaneously the unsteady Navier-Stokes or Euler equations govern-ing the unsteady flows (using finite difference or finite volume formulations, which involve numerousiterations for each real time step) and the structural equations of motion requires a substantially largecomputational effort in terms of computing time and memory, even with the present computing capa-bilities. For this reason, efficient analytical solutions in closed form can be useful in the aeroelasticstudies.

The analytical method presented has been validated for the case of rigid airfoil oscillations inunsteady compressible and incompressible flows. The solutions obtained with this method have beenfound to be in good agreement with the results based on Jordan’s data [51] for oscillating rigid airfoilsin unsteady compressible flows (especially for the case of low oscillation frequencies), and in perfectagreement for the limit case (M → 0) with the results obtained for unsteady incompressible flows byTheodorsen [34], Postel & Leppert [40] and by Mateescu & Abdo [14].

Detailed analytical solutions of the unsteady pressure coefficient distribution on the airfoil havebeen obtained with this method for the case of flexural oscillations in compressible flows. The vari-ations of these coefficients with the Mach number and the reduced frequency of oscillations are alsopresented. The analytical solutions obtained are computationally very efficient and can be success-fully used in the aeroelastic studies,

2 Problem formulation

Consider a thin airfoil of chord c placed in a uniform air flow defined by the velocity U∞ and by

the Mach number M∞ = U∞/

a∞, where a∞ =√

γp∞/

ρ∞ is the speed of sound, in which p∞, ρ∞ andγ stand for the pressure, density and specific heat ratio. This airfoil can execute pitching, plungingand/or flexural oscillations of low frequency, defined as discussed further by appropriate boundaryconditions on the oscillating airfoil.

The unsteady flow past the oscillating airfoil is referred to a Cartesian system of coordinates cx,cy, (where x and y are dimensionless Cartesian coordinates with respect to the chord c), with theorigin at the airfoil leading edge and the axis cx parallel to the direction of the uniform stream. TheCartesian components of the fluid velocity in this unsteady flow are denoted by U∞ (1+u) and U∞ v,

4 Dan Mateescu

where u(x,y, t) = ∂ϕ/

∂x and v(x,y, t) = ∂ϕ/

∂y are the dimensionless components of the perturbationvelocity generated by the oscillating airfoil, which derive from the perturbation velocity potentialdenoted as U∞ cϕ (x,y, t).

Following the same convention as in a previous paper [14] to use a complex function represen-tation for the unsteady flow variables and implying the real part of these complex functions (forconvenience the real part is not specifically written), the harmonic pitching, plunging and/or flexu-ral oscillations of the airfoil about a mean position situated along the axis Ox can be defined in thegeneral complex form

cy = ce (x, t) = ce (x) exp (iω t) , (2.1)

where exp (iω t) = cos ω t + i sin ω t, ω is the radian frequency of oscillations and i =√−1, and

where e(x) represents the modal amplitude of oscillations (usually small) which can be expressed inthe general polynomial form

e (x) =N+1

∑n=0

enxn . (2.2)

In the case of the rigid airfoil oscillations, the modal amplitude of oscillation can be expressed ase(x) = h− (x−a) θ, where h is the complex amplitude of the airfoil oscillations in translation nor-mal to its chord (plunging) defined as h (t) = h exp (iω t), and θ is the complex amplitude of thepitching oscillations about an articulation situated at x = a, which is defined as θ (t) = θ exp (iω t).In the above expressions, the reduced quantities marked by a caret, such as θ, h and e are in generalcomplex numbers defining the amplitude (θA,hA,eA) and the relative phase (ψθ,ψh,ψe) in the formθ = θA exp(iψθ), h = hA exp(iψh) and e = eA exp(iψe).

The boundary condition on this thin airfoil executing small amplitude oscillations can be ex-pressed, as shown in our previous paper [14], in the form

v (x, t) =c

U∞

∂e∂t

+(1+u)∂e∂x

= F (x) exp (iω t) , (2.3)

F (x) = i2k e (x)+∂ e∂x

, k =ω c2U∞

, (2.4)

where k = ω c/(2U∞) is the reduced frequency of oscillations, which is usually small in the aeroelas-

tic oscillations of the aircraft wings.The equation of the perturbation velocity potential for compressible flows can be expressed in the

linear form (Carafoli, Mateescu and Nastase [55] )

(1−M2

∞) ∂2ϕ

∂x2 +∂2ϕ∂y2 =

ca2

∞

[c

∂2ϕ∂ t2 +2U∞

∂2ϕ∂ t ∂x

], (2.5)

and the unsteady pressure coefficient, Cp (x,y, t), is defined by the corresponding linear equation ob-tained from the Bernoulli-Lagrange equation [14, 55]

Cp (x,y, t) =−2u (x,y, t)−(2c

/U∞

) (∂ϕ

/∂ t

). (2.6)

3 Method of solution

By introducing the reduced velocity potential ϕ(x,y) defined by the potential transformation


ϕ (x,y, t) = ϕ (x,y) exp (i2k mx) exp(iω t) , m =M2

∞1−M2

∞, (3.1)

where k is the reduced frequency of oscillations and m is a compressibility parameter, equation (2.5)can be reduced in the case of low frequency oscillations (k2m << 1) to the steady flow form

β2 ∂2ϕ∂x2 +

∂2ϕ∂y2 = 0, β =

√1−M2

∞. (3.2)

The velocity potential transformation (3.1) leads to the reduced perturbation velocity components, uand v, defined as

u (x,y) =∂ϕ∂x

, u(x,y, t) = [u (x,y)+ i2k m ϕ (x,y)] exp (i2k mx) exp(iω t) , (3.3)

v (x,y) =∂ϕ∂y

, v(x,y, t) = v (x,y) exp (i2k mx) exp(iω t) , (3.4)

and the boundary conditions (2.3),(2.4) on the oscillating airfoil can be recast as

v = F (x) exp (− i2k mx) . (3.5)

By using convenient transformations of the coordinates and the reduced potential, in the form

X = x, Y = βy, Φ(X ,Y ) = βϕ(x,y) , (3.6)

equation (3.2) can be reduced to the Laplace equation

∇2Φ =∂2Φ∂X2 +

∂2Φ∂Y 2 = 0, (3.7)

which is also satisfied by the velocity components in this transformed plane

U (X ,Y ) =∂Φ∂X

, u (x,y) =1β

[U (X ,Y )+ i2k mΦ (X ,Y )] exp( i2k mX) , ∇2U = 0, (3.8)

V (X ,Y ) =∂Φ∂Y

, v (x,y) = V (X ,Y ) exp( i2k mX) , ∇2V = 0, (3.9)

Thus, the velocity components are harmonic functions in this transformed plane and can be expressedas the real and imaginary parts of the complex conjugate velocity

w (Z) = f ′ (Z) = U (X ,Y )− j V (X ,Y ) , Z = X + j Y, (3.10)

where f (Z) is the complex potential in this plane (in which j =√−1) defined as

f (Z) =∫

w (Z) dZ, (3.11)

which is related to the dimensionless perturbation velocity potential ϕ (x,y, t) by the equation

Re j { f (Z)}= Φ (X ,Y ) = β ϕ (x,y) = β ϕ (x,y, t) exp (−i2k mx) exp (−iω t) . (3.12)

The boundary condition on the oscillating airfoil can thus be expressed in this plane in the complexform

6 Dan Mateescu

−V (X ,0) = Im j {w(Z)}Z=X =−F (X) , (3.13)

whereF (X) = F (X) exp (−i2k mX) = [e′(X)+ i2k e(X)]exp (− i2k mX) . (3.14)

In the case of low frequency oscillations, F (X)can be expressed as

F (X) =[

e′ (X)+ i2k e(X)] [

1− i2k mX−2k2m2X2] =N+1

∑n=0

fnXn, (3.15)

in which

fn = (n+1) en+1 + i2k (1−nm) en +2k2m [2− (n−1) m] en−1, with en = 0 for n > N. (3.16)

Thus, the unsteady problem in compressible flow is reduced to the solution of an equivalent reducedsteady incompressible flow (with more complex boundary conditions), which can be solved in asimilar manner to that indicated in our previous paper [14] devoted to the solution of unsteady incom-pressible flows.

The boundary condition upstream of the airfoil on the extension of its chord [14] is

U (X ,0) = Re j {w(Z)}Z=X = 0, for Z = X < 0. (3.17)

4 Analysis of the free vortex sheet in the airfoil wake

The circulation around the airfoil contour varies during the oscillatory cycle, and as a result sheddingvortices are generated continuously at the trailing edge (according to Kelvin’s theorem). The inten-sity of these free vortices shed at the trailing edge (x = 1) of the oscillating airfoil at time t can bedetermined from Kelvin’s theorem for a closed material contour in the form [14]

γ f (1, t) =−d ΓC

cdx=− 1

U∞

d ΓC

d t, (4.1)

where ΓC (t) is the circulation around the oscillating airfoil [14] defined as

ΓC (t) = 2cU∞

∫ 1

0u(x,0, t) dx = cU∞ ΓC exp (iω t) , (4.2)

in which the reduced circulation ΓC is defined as

ΓC =2β

∫ 1

0exp ( i2k mX) Re j {w(Z)+ i2k f (Z)}Z=X d X . (4.3)

Based on Helmholtz’ circulation theorem, the free vortices maintain their intensity while they aretransported downstream by the moving fluid, and thus the intensity of a distributed free vortex situatedat time t in the wake at the location cx = cσ is equal to that issued at the trailing edge at a previoustime t−∆ t, where the time lag ∆ t is the time needed for the free vortex to move (with the mean fluidvelocity U∞) from the trailing edge (where it was generated) to the new position, ∆ t = c (σ−1)

/U∞.

Hence,γ f (σ, t) =−i2kU∞ ΓC exp [iω t− i2 (σ−1)] . (4.4)


By considering an elementary contour around an infinitesimal length cdσ of the wake, as shown inour previous paper [14], the distributed free vortices intensity can be expressed as

γ(σ, t) = 2U∞ u(x,0, t) = 2U∞1β

[U (X , 0)+ i2k m Φ (X , 0)] exp (i2k mx) exp(iω t) , (4.5)

and thus the resulting boundary condition on the wake of the oscillating airfoil is

[U (X ,0)+ i2k m Φ (X ,0)]X=σ =−i k βΓC exp (−i2k mX) exp [−i2K (σ−1)] , (4.6)

where K = k (1+m). Since U (X ,0) =(∂/

∂X)

Φ(X ,0), equation (4.6) can be viewed as a differentialequation with the following solution which represents the boundary condition on the wake:

U (σ,0) =−iK βΓC exp (−i2k m) exp [−i2K (σ−1)] , K = k (1+m) . (4.7)

5 Contribution of the free vortex sheet in the expression of w(Z)

The boundary conditions for the contribution ΓC ww (Z) of the free vortices in the wake defined by thereduced circulation ΓC around the airfoil, can be expressed, taking into account (4.4) and (4.7), in thecomplex form

Re j{

ΓC ww (Z)}

Z=X =−iK βΓC exp (−i2k m) exp [−i2K (σ−1)] for Z = X > 1,

Re j{

ΓC ww (Z)}

Z=X = 0 for Z = X < 0, (5.1)

Im j{

ΓC ww (Z)}

Z=X = 0 for Z = X ∈ [0, 1] .

The solution of this problem, ΓC ww (Z), can be obtained in a similar manner to that presented in ourprevious paper [14] in the form

ww (Z) =−12

β exp (−i2k m)∫ ∞

1exp [−i2K (σ−1)] H (Z,σ) dσ, (5.2)

where

H (Z,σ) =2π

cos−1

√(1−Z) σ

σ−Z. (5.3)

6 Basic unsteady flow solution δw(Z)

Consider first the basic unsteady flow problem corresponding to a sudden change δF0 in the boundarycondition at Z = S in the complex plane Z, which is defined by boundary conditions including aHeaviside step function in the form

Im j {δw(Z)}Z=X =− f0 for Z = X ∈ [0, S] ,Im j {δw(Z)}Z=X =− f0 +δF0 for Z = X ∈ [S, 1] ,

Re j {δw(Z)}Z=X =−iK β δΓC exp (−i2k m) exp [−i2K (σ−1)] for Z = X > 1, (6.1)

Re j {δw(Z)}Z=X = 0 for Z = X < 0,

8 Dan Mateescu

where δF0 defines a jump in the airfoil boundary condition located at the ridge Z = X = S, and δΓC

represents the reduced circulation around the airfoil in this case. The solution for this problem can beexpressed in a similar form to that obtained in our previous paper [14]

δw(Z) =− j f0 +δA

√1−Z

Z−δF0 G(Z,S)+δΓC ww (Z) , (6.2)

in which G(Z,S) represents the contribution of the ridge situated at Z = S (where the boundarycondition changes) defined as

G(Z,S) =2π

cosh−1

√(1−Z) S

S−Z. (6.3)

The constant δA can be determined from the condition at infinity where the perturbation velocitycomponents become zero [14] resulting

δA =− f0−δF0C (S)+2K2 β δΓC exp (−i k m)∫ ∞

1exp [ i2K (σ−1)]

2π

cosh−1√σ dσ , (6.4)

whereC (S) =

2π

cos−1√

S , (6.5)

The solution for this basic unsteady problem (corresponding to the change δF0 in the boundary con-dition at Z = S in the complex plane Z) can thus be expressed as

δw(Z) =− f0

(√1−Z

Z+ j

)−δF0

[G(Z,S)−C (S)

√1−Z

Z

]− iK β δΓC exp(−i2k m) J (Z) ,

(6.6)where

J (Z) =∫ ∞

1exp [ i2K (σ−1)]

1π

1σ−Z

√1−Z

Z

√σ

σ−1dσ . (6.7)

The solution for δ f (Z) =∫

δw (Z) dZ, defined by an equation similar to (3.11), is then obtained inthe form

δ f (Z) =− [ f0 +δF0C (S)][−π

2C (Z)+

√(1−Z)Z

]− j f0 Z

−δF0

[(Z−S) G(Z,S)−

√(1−S)S C (Z)

]− iK β δΓC exp (−i2k m) J (Z) , (6.8)

where

J (Z) =∫ ∞

1exp [ i2K (σ−1)]

[−H (Z,σ)−

√σ

σ−1C (Z)

]dσ . (6.9)

Determination of the reduced circulation δ ΓC

The reduced circulation for the prototype unsteady problem is defined, similarly to (4.3), as

δΓC =2β

∫ 1

0exp ( i2k mX) Re j {δw(Z)+ i2k δ f (Z)}Z=X d X , (6.10)

which leads, after performing the calculations for the case of low frequency oscillations, to the ex-pression


δΓC =πβ

i exp(i2k m)2K B1 (K)

[f0 +δF0C (S)+δF0

2π

√(1−S)S

], (6.11)

whereB1 (K) =−π

4exp ( iK)

[H(2)

1 (K)+ iH(2)0 (K)

]

=−π4

exp ( iK) { [J1 (K)+Y0 (K)]+ i [J0 (K)−Y1 (K)]} , (6.12)

in which H(2)n (K) denotes the Hankel functions of second kind and of order n, and Jn (K) and Yn (K)

are, respectively, the Bessel functions of the first and second kind and order n.

Reduced pressure coefficient δCp (x)

The unsteady pressure coefficient for the flow past the oscillating airfoil can be expressed, using (2.6),(3.3), (3.8), (3.10) and (3.12), in the form

Cp (x,y, t) = Cp (x,y) exp ( iω t) , (6.13)

where Cp (x,y) is the reduced pressure coefficient defined as

Cp (x,y) =−2β

exp ( i2k mX) Re j {g(Z)}Z=X , g(z) = w(Z)+ i2K f (Z) . (6.14)

For the basic unsteady flow problem, one results

δg(Z) =− [ f0 +δF0C (S)] [1+ i2K Z +D(K) ]

√1−Z

Z− j f0 (1+ i2K Z)

−δF0

{[1+ i2K (Z−S)] G(Z,S)+

2π

D(K)√

(1−S)S

√1−Z

Z

}, (6.15)

and hence

δCp (x) =2β

exp (i2k mx)

{[ f0 +δF0C (S) ] [1+ i2K x+D(K) ]

√1− x

x

+δF0

{[1+ i2K (x−S) ] G(x,S)+

2π

√(1−S)SD(K)

√1− x

x

}}, (6.16)

where

D(K) =−iH(2)

0 (K)

H(2)1 (K)+ iH(2)

0 (K)= CT (K)−1, K =

k1−M2

∞, (6.17)

G(x,S) = Re j{

G(Z,S)}

=

2π cosh−1

√(1−x)S

S−x f or x ∈ [0,S]2π sinh−1

√(1−x)S

x−S f or x ∈ [S,1]

0 f or x < 0 and x > 1

, (6.18)

in which CT (K) denotes here the Theodorsen function, but of a different argument than in incom-pressible flows, K = k (1+m) = k

/(1−M2

∞).

10 Dan Mateescu

0.0 0.2 0.4 0.6 0.8 1.0-3

0

3

M=0: Present solutionM=0.3: Present solutionM=0.4: Present solutionM=0.5: Present solutionM=0.6: Present solutionM=0: Postel & LeppertM=0: Mateescu & Abdo

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16


k=0.24

k=0.24

x x

Real

ˆ ˆpC !

Imag

ˆ ˆpC !

Fig. 1 Variations with the Mach number of the chordwise distributions of the real and imaginary components of thereduced pressure difference coefficient, ∆Cp(x)

/θ , for a thin airfoil executing pitching oscillations, θ(t) = θexp(iωt),

about an articulation located at a = 0.25, for k = 0.24. For M = 0, comparison between: Present solutions (−), Postel &Leppert [40] results (•), and Mateescu & Abdo [14] solutions (¤).

Lift and pitching moment coefficients for the basic unsteady flow

The lift and pitching moment (with respect to the leading edge) coefficients for the oscillating airfoilare defined as

CL (t) = CL exp ( iω t) , CL =−2∫ 1

0Cp (x) dx, (6.19)

Cm (t) = Cm exp ( iω t) , Cm =−2∫ 1

0xCp (x) dx, (6.20)

For the basic unsteady flow problem, the lift and pitching moment coefficients are defined as

δCL =−2∫ 1

0δCp (x) dx , δCm =−2

∫ 1

0xδCp (x) dx, (6.21)

and one obtains the following expressions after performing the integration

δCL =−2πβ

[f0 +δF0C (S)+δF0

2π

√(1−S)S

] {1+D(K)+ i

k2

[1+2m+mD(K)]−

−m4

k2 [2+3m+mD(K)]}

+4β

δF0√

(1−S)S[i k +

m3

k2 (2+3m) S (1+2S )]

, (6.22)

δCm =− π2β

[f0 +δF0C (S)+δF0

2π

√(1−S)S

] {1+D(K)+ i k + i k m [2+D(K)]− 5

8k2m [2+3m+

+mD(K)]}− 1

βδF0

√(1−S)SS

{2− i k

23

(1−m) (1−2S)− k2m (2+3m)(

34

+S +2S2)}

,

(6.23)


k=0.10

k=0.10

0.0 0.2 0.4 0.6 0.8 1.0-6

-4

-2

0

M=0: Present solutionM=0.3: Present solutionM=0.4: Present solutionM=0.5: Present solutionM=0.6: Present solutionM=0.7: Present solutionM=0: Mateescu & Abdo

0.0 0.2 0.4 0.6 0.8 1.00

8

16

24


k=0.05

k=0.05

Real

ˆ ˆpC !

Imag

ˆ ˆpC !

xx

0.0 0.2 0.4 0.6 0.8 1.0-6

-4

-2

0


0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16


k=0.10

k=0.10Real

ˆ ˆpC !

Imag

ˆ ˆpC !

x x

Fig. 2 Typical variations with the Mach number of the chordwise distributions of the real and imaginary components of thereduced pressure difference coefficient, ∆Cp(x)

/θ , for a thin airfoil executing pitching oscillations, θ (t) = θ exp (iω t),

about an articulation located at a = 0.25, for two values of the reduced frequency of oscillations, k = 0.05 and k = 0.10.For M = 0, comparison between: Present solutions (−), and Mateescu & Abdo [14] solutions (¤).

7 General solution of the complete unsteady flow problem

The solution of the complete compressible flow problem of an oscillating airfoil can be obtainedby considering a continuous distribution of elementary ridges along the airfoil chord to model theboundary conditions (3.13) – (3.15). For each of these elementary ridges situated at Z = S, the changeδF0 in the boundary conditions is thus

δF0 =[

d F (X)d X

]

X=SdS = F ′ (S) dS =

N+1

∑n=0

n fn Sn−1 dS , (7.1)

where the coefficients fn are defined by (20) in function of the modal oscillation amplitude e(x).

12 Dan Mateescu

0.0 0.2 0.4 0.6 0.8 1.0-4

-2

0

2

M=0: Present solutionM=0.3: Present solutionM=0.4: Present solutionM=0.5: Present solutionM=0.6: Present solutionM=0: Mateescu & Abdo

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16


x x

Real

ˆ ˆpC !

k=0.15

k=0.15Imag

ˆ ˆpC !

0.0 0.2 0.4 0.6 0.8 1.0-4

-2

0

2


0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16


x x

Real

ˆ ˆpC !

Imag

ˆ ˆpC !

k=0.20

k=0.20

Fig. 3 Typical variations with the Mach number of the chordwise distributions of the real and imaginary components of thereduced pressure difference coefficient, ∆Cp(x)

/θ , for a thin airfoil executing pitching oscillations, θ (t) = θ exp (iω t),

about an articulation located at a = 0.25, for two values of the reduced frequency of oscillations, k = 0.15 and k = 0.20.For M = 0, comparison between: Present solutions (−), and Mateescu & Abdo [14] solutions (¤).

By introducing the expression (7.1) of δF0 in the basic unsteady solutions (6.16),(6.22) and(6.23)for the basic pressure difference, lift and moment coefficients, and performing the integration in Sbetween S = 0 and S = 1, one obtains the general solutions of the unsteady pressure difference coef-ficient between the lower and upper sides of the oscillating airfoil,

∆Cp (x, t) =−2Cp (x, t) = ∆Cp (x) exp ( iω t) , (7.2)

as well as the unsteady lift and pitching moment (with respect to the leading edge) coefficients,

CL ( t) = CL exp ( iω t) , Cm ( t) = Cm exp ( iω t) . (7.3)

where the resulting general expressions for the reduced coefficients ∆Cp (x), CL and Cm are


0 0.1 0.2 0.3

1

2

3

4

M=0: Present solutionM=0.3: Present solutionM=0.4: Present solutionM=0.5: Present solutionM=0.6: Present solutionM=0.7: Present solutionM=0: TheodorsenM=0: Mateescu & Abdo

0 0.1 0.2 0.3-1

0

1

2

3


k k

Realˆ ˆmC

Imagˆ ˆmC

0 0.1 0.2 0.34

6

8

10

12


0 0.1 0.2 0.3-3

0

3

6

9


k k

Real

ˆ ˆLC

Imag

ˆ ˆLC

Fig. 4 Typical variations with the reduced frequency of oscillations, k, and the Mach number, M, of the real and imaginarycomponents of the reduced lift and pitching moment coefficients, CL

/θ and Cm

/θ , for a thin airfoil executing pitching

oscillations, θ (t) = θ exp (iω t), with respect to the leading edge (a = 0). For M = 0, comparison between: Present solutions(−), Theodorsen [34] results (•) and Mateescu & Abdo [14] solutions (¤).

∆Cp (x) =−4β

exp (i2k mx)

√1− x

x

N+1

∑n=0

fn

{2n+1n+1

gn D(K)+n

∑q=0

gn−q

[1+

i2k (1+m)n+1

x]

xq

},

(7.4)

CL =−2πβ

N+1

∑n=0

fn gn2n+1n+1

{1+

i kn+2

+ imk +[1+

m4

k (2 i−mk)]

D(K)− 34

mk2 (2+3m)n+1n+3

},

(7.5)

Cm =− π2β

N+1

∑n=0

fn gn2n+1n+1

{2n+2n+2

+ i3k

n+3[1+m (n+2)]+

[1+ imk− 5

8m2k2

]D(K)

14 Dan Mateescu

0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8


0.0 0.2 0.4 0.6 0.8 1.0-1

0

1

2

3


k=0.24

k=0.24

x x

Real

ˆˆpC h !

Imag

ˆˆpC h !

Fig. 5 Variations with the Mach number of the chordwise distributions of the real and imaginary components of thereduced pressure difference coefficient, ∆Cp(x)

/h , for a thin airfoil executing normal-to-chord oscillatory translations,

h (t) = h exp (iω t), for k = 0.24.For M = 0, comparison between: Present solutions (−), Postel & Leppert [40] results (•),and Mateescu & Abdo [14] solutions (¤).

−5mk2(

1+32

m)

n+1n+4

}. (7.6)

In these expressions, fn are defined by Eq. (3.16) in function of the coefficients en of the modal ampli-tude of oscillations, e(x), D(K) of K = k (1+m) is defined by Eq. (6.17), and gn =(2n) !

/[22n (n !)2

]

is also defined by the recurrence formula

gn = gn−1 (2n−1)/(2n), g0 = 1, g1 = 1

/2 . (7.7)

Thus, simple algebraic expressions in closed form (7.4) - (7.6) have been obtained for the unsteadyaerodynamic coefficients in the general case of oscillating rigid or flexible airfoils in compressibleflows. These general expressions of the reduced coefficients ∆Cp (x), CL and Cm are also valid inthe limit case of incompressible flows (m = 0, β = 1 and K = k) when they become identical to thecorresponding expressions derived for incompressible flows in our previous paper, Mateescu & Abdo[14].

The actual time variations of the pressure difference, lift and pitching moment coefficients,∆Cp (x, t), CL (t) and Cm (t), can be calculated by taking the real parts of the complex expressions(7.2) and (7.3) in the form

∆Cp (x, t) = Re{

∆Cp (x)}

cos(ω t)− Im{

∆Cp (x)}

sin(ω t) , (7.8)

CL (t) = Re{

CL}

cos(ω t)− Im{

CL}

sin(ω t) , (7.9)

Cm (t) = Re{

Cm}

cos(ω t)− Im{

Cm}

sin(ω t) , (7.10)

In this case, the actual time variations of the airfoil positions in oscillatory pitching rotation andnormal-to-chord translation can be expressed in the form

θ (t) = θA cos(ω t +ψθ) , h (t) = hA cos(ω t +ψh) , (7.11)

where ψθ and ψh define the relative phase of these oscillatory motions.


0.0 0.2 0.4 0.6 0.8 1.0

0

1

2


0.0 0.2 0.4 0.6 0.8 1.00

1

2

3

4


k=0.10 k=0.10

x x

Real

ˆˆpC h !

Imag

ˆˆpC h !

0.0 0.2 0.4 0.6 0.8 1.00

1

2


0.0 0.2 0.4 0.6 0.8 1.00

1

2

3

4


k=0.05

k=0.05

x x

Real

ˆˆpC h !

Imag

ˆˆpC h !

Fig. 6 Typical variations with the Mach number of the chordwise distributions of the real and imaginary components ofthe reduced pressure difference coefficient, ∆Cp(x)


h (t) = h exp (iω t), for two values of the reduced frequency of oscillations, k = 0.05 and k = 0.10. For M = 0, comparisonbetween: Present solutions (−), and Mateescu & Abdo [14] solutions (¤).

8 Results for pitching and plunging oscillations of rigid airfoils and method validation

For rigid airfoil oscillations in pitching rotation, θ (t) = θ exp (iω t), about an articulation situated atx = a, and in normal-to-chord translation, h (t) = h exp (iω t), the modal amplitude of oscillation is

e(x) = h− (x−a) θ, e0 = h+a θ, e1 =−θ, en = 0 f or n > 1, (8.1)

and hence the values of the coefficients fn appearing in the solutions (7.4 ) - (7.6) are in this case

f0 = e1 + i2k e0 =− θ+ i2k(h+a θ

), f2 = 2k2m [2− m] e1 =−2k2m [2− m] θ, (8.2)

16 Dan Mateescu

0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8


0.0 0.2 0.4 0.6 0.8 1.0-1

0

1

2

3


k=0.20

k=0.20

x x

Real

ˆˆpC h !

Imag

ˆˆpC h !

0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8


0.0 0.2 0.4 0.6 0.8 1.0-1

0

1

2

3


k=0.15

k=0.15

x x

Real

ˆˆpC h !

Imag

ˆˆpC h !



h (t) = h exp (iω t), for two values of the reduced frequency of oscillations, k = 0.15 and k = 0.20. For M = 0, comparisonbetween: Present solutions (−), and Mateescu & Abdo [14] solutions (¤).

f1 = i2k (1−m) e1 +4k2me0 =−i2k (1−m) θ+4k2m(h+a θ

), fn = 0 f or n > 2. (8.3)

These values of the coefficients fn are inserted in equations (7.4) – (7.6) in order to obtain the specificsolutions of the unsteady pressure difference, lift and pitching moment (with respect to the leadingedge) coefficients for the case of the rigid airfoil oscillations. The specific values of these unsteadyaerodynamic coefficients are presented in the following for various Mach numbers and reduced fre-quencies of the oscillations.

One may note that for the same radian frequency of oscillation, ω, of an airplane (which dependson its mass and moment of inertia), the reduced frequency of oscillations, k = ω c

/(2U∞), is inverse

proportional to the Mach number, that is k = kin(M∞ in

/M∞

) (a∞ in

/a∞

), where the subscript “in”

denotes the values corresponding to the incompressible flow case. This relation, which is consistent


0 0.1 0.2 0.30

2

4

6


0 0.1 0.2 0.3-1.5

-1.0

-0.5

0.0

0.5


k k

Real

ˆˆLC h

Imag

ˆˆLC h

0 0.1 0.2 0.3-1.0

-0.5

0.0

0.5


0 0.1 0.2 0.30.0

0.5

1.0

1.5

2.0


k k

Real

ˆˆmC h

Imag

ˆˆmC h

Fig. 8 Typical variations with the reduced frequency of oscillations, k, and the Mach number, M, of the real and imaginarycomponents of the reduced lift and pitching moment coefficients, CL

/h and Cm

/h , for a thin airfoil executing normal-to-

chord oscillatory translations, h (t) = h exp (iω t). For M = 0, comparison between: Present solutions (−), Theodorsen [34]results (•) and Mateescu & Abdo [14] solutions (¤).

with the assumption of low frequency oscillations, k2M2∞/(

1−M2∞)

<< 1, is considered in definingthe values of M∞ and k for which the unsteady aerodynamic coefficients are presented in the followingfigures. In these figures, M∞ is replaced by M for convenience.

8.1 Solutions for pitching oscillations (h = 0)

The typical influence of the Mach number, M, and of the reduced oscillation frequency, k, on thechordwise distributions of the real and imaginary components of the pressure difference coefficient(between the lower and the upper surfaces of the airfoil), ∆Cp (x)

/θ, are shown in Figures 1 to 3

18 Dan Mateescu

0 0.2 0.4 0.61

2

3

k=0.00000001: Present solutionk=0.05: Present solutionk=0.10: Present solutionk=0.15: Present solutionk=0.20: Present solutionTheodorsen solutionMateescu & Abdo solutionk=0: Prandtl-Glauert (steady)

0 0.2 0.4 0.64

8

12

k=0.00000001: Present solutionk=0.05: Present solutionk=0.10: Present solutionk=0.15: Present solutionk=0.20: Present solutionTheodorsen solutionMateescu & Abdo solutionk=0: Prandtl-Glauert (steady)

Real

ˆ ˆLC

Realˆ ˆmC

M M

Fig. 9 Variations with the Mach number, M, and the reduced frequency of oscillations, k, of the real components ofthe reduced lift and pitching moment coefficients, CL

/θ and Cm

/θ , for a thin airfoil executing pitching oscillations,

θ (t) = θ exp (iω t), with respect to the leading edge (a = 0). Comparison between: Present solutions (−), Theodorsen [34]results for M = 0 (•), Mateescu & Abdo [14] solutions for M = 0 (¤), and Solutions based on Prandtl-Glauert similarityrule for steady compressible flows, which corresponds to k → 0 (©).

for the case of pitching oscillations, θ (t) = θ exp (iω t), about an articulation located at a distanceca = 0.25c from the leading edge.

The solutions for the pressure difference coefficient obtained with the present method for the limitcase of incompressible flows (M → 0) were found to be in excellent agreement with the previousresults obtained for unsteady incompressible flows by Postel and Leppert [40] and by Mateescu andAbdo [14], as shown in Figures 1 to 3.

The variations with the reduced frequency of oscillation, k = ω c/(2U∞), for various Mach num-

bers, M, of the real and imaginary components of the reduced lift and pitching moment (with respectto the leading edge) coefficients, CL

/θ and Cm

/θ, are illustrated in Figure 4 for the case of pitching

oscillations with respect to the leading edge (a = 0) defined by θ (t) = θ exp (iω t).In all cases, the solutions obtained with the present method for the limit case of incompressible

flows (M → 0) were found to be in excellent agreement with the previous incompressible flow solu-tions obtained by Theodorsen [34] and by Mateescu and Abdo [14].

8.2 Solutions for normal-to-chord oscillatory translation (θ = 0)

The typical influence of the Mach number, M, and of the reduced oscillation frequency, k, on thechordwise distributions of the real and imaginary components of the pressure difference coefficient,∆Cp (x)

/h, are shown in Figures 5 to 7 for the case of normal-to-chord oscillatory translations, defined

by h (t) = h exp (iω t).The solutions for the pressure difference coefficient obtained with the present method for the limit

case of incompressible flows (M → 0) were found to be in excellent agreement with the previousresults obtained for unsteady incompressible flows by Postel and Leppert [40] and by Mateescu andAbdo [14], as shown in Figures 5 to 7.


A

~-1.0 -0.5 0.0 0.5 1.0-8

-4

0

4

8Present theoryResults based on Jordan' s data

-1.0 -0.5 0.0 0.5 1.0-8

-4

0

4

8Present theoryTheodorsen solutionMateescu & Abdo solution

ALC ~

ALC ~

AmC ~

AmC ~

A

~

ALC ~

AmC ~

k=0.05

M=0

ALC ~

AmC ~

k=0.05

M=0.5

-1.0 -0.5 0.0 0.5 1.0-8

-4

0

4


-1.0 -0.5 0.0 0.5 1.0-8

-4

0

4


ALC ~

AmC ~

k=0.05

M=0.6

ALC ~

AmC ~

k=0.05

M=0.7

ALC ~

ALC ~

AmC ~

AmC ~

A

~A

~

Fig. 10 Typical variations with the airfoil position during the oscillatory cycle, θ(t)/

θA , of the unsteady lift and pitchingmoment coefficients, CL(t)

/θA and Cm(t)

/θA , for the case of pitching oscillations, θ (t) = θA cos(ω t), with respect to the

leading edge (a = 0), for various Mach numbers (M = 0 , 0.5 , 0.6 and 0.7) and for the reduced frequency of oscillationsk = 0.05. Comparison between the present solutions (−) and the results based on Jordan’s data [51] (©), as well as withthe results based on the solutions derived for M = 0 by Theodorsen [34] (•) and by Mateescu & Abdo [14] (¤).

The variations with the reduced frequency of oscillation, k = ω c/(2U∞), for various Mach num-

bers, M, of the real and imaginary components of the reduced lift and pitching moment (with respectto the leading edge) coefficients, CL

/h and Cm

/h, are illustrated in Figure 8 for the case of of normal-

to-chord oscillatory translations defined by h (t) = h exp (iω t).In all cases, the solutions obtained with the present method for the limit case of incompressible

flows (M → 0) were found to be in excellent agreement with the previous incompressible flow solu-tions obtained by Theodorsen [34] and by Mateescu and Abdo [14].

20 Dan Mateescu

-1.0 -0.5 0.0 0.5 1.0-8

-4

0

4


-1.0 -0.5 0.0 0.5 1.0-8

-4

0

4


ALC ~

AmC ~

k=0.05

M=0.5

ALC ~

AmC ~

k=0.10

M=0.5

ALC ~

ALC ~

AmC ~

AmC ~

A

~A

~

-1.0 -0.5 0.0 0.5 1.0-8

-4

0

4


-1.0 -0.5 0.0 0.5 1.0-8

-4

0

4


ALC ~

AmC ~

k=0.15

M=0.5

ALC ~

AmC ~

k=0.20

M=0.5

ALC ~

ALC ~

AmC ~

AmC ~

A

~A

~



/θA and Cm(t)

/θA , for the case of pitching oscillations, θ (t) = θA cos(ω t), with respect to

the leading edge (a = 0), for Mach number M = 0.5 and for various values of the reduced frequencies of oscillations,k = 0.05 , 0.10 , 0.15 and 0.20. Comparison between the present solutions (−) and the results based on Jordan’s data [51](©).

8.3 Method validation

As shown in Figures 1 to 8, the present solutions for the unsteady pressure distribution, unsteady liftand pitching moment coefficients obtained in the limit case of incompressible flow (M → 0) are inexcellent agreement with those obtained by Theodorsen [34], Postel & Leppert [40] and by Mateescu& Abdo [14] for incompressible flows.

Also, in the limit case of vanishingly small oscillation frequency (k → 0), the present analyticsolutions are in excellent agreement with the solutions for steady compressible flows based on thePrandtl-Glauert similarity rule, as shown in Figure 9 for the real components of the reduced lift


-1.0 -0.5 0.0 0.5 1.0-1

0

1

2

Present theoryResults based on Jordan' s data

-1.0 -0.5 0.0 0.5 1.0-1

0

1

2

Present theoryTheodorsen solutionMateescu & Abdo solution

ALhC

~

AmhC

~

ALhC

~

AmhC

~

ALhC

~

AmhC

~

k=0.05

M=0

ALhC

~

AmhC

~

k=0.05

M=0.5

Ahh

~Ahh

~

-1.0 -0.5 0.0 0.5 1.0-1

0

1

2


-1.0 -0.5 0.0 0.5 1.0-1

0

1

2


ALhC

~

AmhC

~

ALhC

~

AmhC

~

ALhC

~

AmhC

~

k=0.05

M=0.6

ALhC

~

AmhC

~

k=0.05

M=0.7

Ahh

~Ahh

~



/θA and Cm(t)

/θA , for the case of normal-to-chord oscillatory translations, h (t) = hA cos(ω t),

for various Mach numbers (M = 0 , 0.5 , 0.6 and 0.7) and for the reduced frequency of oscillations k = 0.05. Comparisonbetween the present solutions (−) and the results based on Jordan’s data [51] ( © ), as well as with the results based on thesolutions derived for M = 0 by Theodorsen [34] (•) and by Mateescu & Abdo [14] (¤).

and pitching moment coefficients, CL/

θ and Cm/

θ, in case of pitching oscillations (the imaginarycomponents are zero for k → 0).

For unsteady compressible flows, the present solutions are compared for validation with the pre-vious results given by Jordan [51], which are based on the procedure presented in [52] for M = 0.7.This is a rather complicated procedure [51] which uses a Fourier series expansions (similar to Kuss-ner [37, 38] ) and requires the approximate solutions of a succession of integral equations with kernelK (s,0) from the incompressible flow problem; it also uses an approximating 9th order polynomialwith coefficients determined separately for each Mach number by fitting the polynomial to tabulatedvalues, as well as the numerical evaluations of several integrals.

The comparison with previous unsteady compressible flow results is shown in Figures 10 to 13,for the variations of the unsteady lift and pitching moment coefficients (with respect to leading edge),

22 Dan Mateescu

-1.0 -0.5 0.0 0.5 1.0-2

0

2


-1.0 -0.5 0.0 0.5 1.0-1

0

1

2


ALhC

~

AmhC

~

ALhC

~

AmhC

~

ALhC

~

AmhC

~

k=0.05

M=0.5

ALhC

~

AmhC

~

k=0.10

M=0.5

Ahh

~Ahh

~

-1.0 -0.5 0.0 0.5 1.0

-2

0

2

4


-1.0 -0.5 0.0 0.5 1.0-2

0

2


ALhC

~

AmhC

~

ALhC

~

AmhC

~

ALhC

~

AmhC

~

k=0.15

M=0.5

ALhC

~

AmhC

~

k=0.20

M=0.5

Ahh

~Ahh

~



/θA and Cm(t)

/θA , for the case of normal-to-chord oscillatory translations, h (t) = hA cos(ω t),

for Mach number M = 0.5 and for various values of the reduced frequencies of oscillations, k = 0.05 , 0.10 , 0.15 and 0.20.Comparison between the present solutions (−) and the results based on Jordan’s data [51] (©).

CL (t)= Re{

CL}

cos(ω t)−Im{

CL}

sin(ω t) and Cm (t)= Re{

Cm}

cos(ω t)−Im{

Cm}

sin(ω t),with the airfoil position during the oscillatory cycle.

Validation for pitching oscillations and normal-to-chord oscillatory translations

The influence of the Mach number, M, on the variations with the airfoil position of the unsteadylift and pitching moment coefficients, is illustrated in Figure 10 for the case of pitching oscillations,θ (t) = θA cos(ω t), with respect to the leading edge (a = 0), and in Figure 12 for the case of normal-to-chord oscillatory translations, h (t) = hA cos(ω t), for various values of the reduced frequencies ofoscillations, k.


0.0 0.2 0.4 0.6 0.8 1.0-12

-8

-4

0


0.0 0.2 0.4 0.6 0.8 1.00

6

12

18

24


k=0.05

k=0.05

x x

Real

2ˆpC e !

Imag

2ˆpC e !

0.0 0.2 0.4 0.6 0.8 1.0-12

-8

-4

0


0.0 0.2 0.4 0.6 0.8 1.00

6

12

18

24


k=0.10

k=0.10

x x

Real

2ˆpC e !

Imag

2ˆpC e !


/e2 , for a thin airfoil executing parabolic flexural oscillations, e(x, t) =

e2 x2 exp(iωt), for two values of the reduced frequency of oscillations, k = 0.05 and k = 0.10. For M = 0, comparisonbetween: Present solutions (−), and Mateescu & Abdo [14] solutions (¤).

The influence of the reduced frequency of oscillation, k, on the variations with the airfoil positionof the unsteady lift and pitching moment coefficients, is illustrated in Figure 11 for the case of pitchingoscillations, θ (t) = θA cos(ω t), with respect to the leading edge (a = 0), and in Figure 13 for the caseof normal-to-chord oscillatory translations, h (t) = hA cos(ω t), for various Mach numbers.

One can notice that in all cases, the present solutions are in good agreement with the results forunsteady compressible flows based on Jordan’s data [51] (especially for low oscillation frequencies),and in excellent agreement in the limit case of incompressible flows (M → 0) with the solutionsobtained for incompressible flows by Theodorsen [34], Postel & Leppert [40] and Mateescu & Abdo[14].

24 Dan Mateescu

0.0 0.2 0.4 0.6 0.8 1.0-8

-4

0


0.0 0.2 0.4 0.6 0.8 1.00

6

12

18

24


k=0.15

k=0.15

x x

Real

2ˆpC e !

Imag

2ˆpC e !

0.0 0.2 0.4 0.6 0.8 1.0-8

-4

0


0.0 0.2 0.4 0.6 0.8 1.00

6

12

18

24


k=0.20

k=0.20

x x

Real

2ˆpC e !

Imag

2ˆpC e !


/e2 , for a thin airfoil executing parabolic flexural oscillations, e(x, t) =

e2 x2 exp ( iω t), for two values of the reduced frequency of oscillations, k = 0.15 and k = 0.20. For M = 0, comparisonbetween: Present solutions (−), and Mateescu & Abdo [14] solutions (¤).

9 Results for flexural oscillations of flexible airfoils

Examples of results are presented for the case of parabolic flexural oscillations defined by

e(x, t) = e2 x2 exp ( iω t) , e(x) = e2 x2, en = 0 f or n 6= 2. (9.1)

In this case, the values of the coefficients fn defined by equation (3.16) and appearing in the solutions(7.4 ) - (7.6) are

f0 = 0, f1 = 2e2, f2 = i2k (1−2m) e2, f3 = 4k2m [1− m] e2, fn = 0 f or n > 3. (9.2)

The variations with the Mach number of the chordwise distributions of the real and imaginary com-ponents of the reduced pressure difference coefficients, ∆Cp (x)

/e2, for the case of parabolic flexural


oscillations are illustrated in Figures 14 and 15 for various values of the reduced frequency of os-cillations. The solutions obtained with the present method for the limit case of incompressible flows(M → 0) were found to be in excellent agreement with the previous solutions obtained by Mateescuand Abdo [14] for incompressible flows (no data were found for comparison in the case of the flexuraloscillations of flexible airfoils in compressible flows).

10 Conclusions

Efficient analytical solutions are presented in this paper for unsteady subsonic compressible flows pastflexible airfoils executing low frequency flexural oscillations. These analytical solutions are obtainedwith a method using velocity singularities related to the airfoil leading edges and the ridges (definedat the locations where the boundary conditions changes), in contrast to Theodorsen’s method forincompressible flows which uses singularities in the expression of the velocity potential.

Simple and efficient analytical solutions in closed form (in contrast with very complicated previoussolutions based on Fourier series expansions, approximate polynomial fitting for each Mach numberand on the numerical evaluations of several integrals) are presented for the chordwise distribution ofthe unsteady pressure difference coefficient and for the unsteady lift and pitching moment coefficientsin the general case of rigid airfoil oscillations. A detailed study of the variations of the unsteadyaerodynamic coefficients with the Mach number and with the reduced frequency of oscillations isalso presented.

These solutions have been found to be in good agreement with the results based on Jordan’s datafor oscillating rigid airfoils in unsteady compressible flows, especially for low oscillation frequencies,and in perfect agreement for the limit case M→ 0 with the solutions of unsteady incompressible flowsobtained by Theodorsen, Postel & Leppert and by Mateescu & Abdo.

Simple and efficient analytical solutions in closed form have also been obtained for the generalcase of flexural oscillations of flexible airfoils. These solutions can be efficiently used in the study ofthe aeroelastic interaction problems requiring the simultaneous solution of the unsteady flow and thestructural deformation motion.

Acknowledgments

The support of the Natural Sciences and Engineering Research Council of Canada is gratefully ac-knowledged, as well as the contribution to the numerical results of a former graduate student, SilviuNeculita.

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Theoretical solutions for unsteady compressible … · Theoretical solutions for unsteady compressible subsonic ... (with applications in aeroelasticity) ... Fung [41], Bisplinghoff