On the high-frequency response of unsteady lift and circulation: A dynamical systems perspective

Journal of Fluids and Structures 93 (2020) 102868

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On the high-frequency response of unsteady lift and circulation: A dynamical systems perspective ∗

Haithem E. Taha , Amir S. Rezaei Mechanical and Aerospace Engineering, University of California, Irvine, CA 92697, United States of America

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Article history: Received 29 August 2019 Received in revised form 21 November 2019 Accepted 2 January 2020 Available online xxxx Keywords: Unsteady aerodynamics Theodorsen Wagner Circulatory and non-circulatory Lift Impulsive start Initial lift High-frequency gain

a b s t r a c t In this paper, we consider the unsteady aerodynamics of a two-dimensional airfoil as a dynamical system whose input is the angle of attack (or airfoil motion) and output is the lift force. Based on this view, we discuss the evolution of lift and circulation from a purely dynamical perspective through step response, frequency response, transfer function, etc. In particular, we point to the relation between the high-frequency gain of the transfer function and the physics of the development of lift and circulation. Based on this view, we show that the circulatory lift dynamics is different from the circulation dynamics. That is, we show that the circulatory lift is not lift due to circulation. In fact, we show that the circulatory–non-circulatory classification is arbitrary. By comparing the steady and unsteady thin airfoil theory, we show that the circulatory lift possesses some acceleration (added-mass) effects. Finally, we perform simulations of Navier– Stokes equations to show that a non-circulatory maneuver in the absence of a free stream induces viscous circulation over the airfoil. © 2020 Elsevier Ltd. All rights reserved.

1. Introduction Interests in unsteady aerodynamics have been continuously increasing over the last century with a recent research flurry because of the modern applications of rapidly maneuvering fighters, highly flexible airplanes, bio-inspired flight, etc. The classical theory of unsteady aerodynamics, whose foundation was laid down by Prandtl (1924) and Birnbaum (1924) in 1924, was mainly set for incompressible, slightly viscous flows around thin airfoils with sharp trailing edges. The key concept is that the flow unsteadiness leads to vorticity generation that emanates at the sharp trailing edge and freely shed behind the airfoil. In addition, the flow outside these sheets can be considered inviscid (e.g., circulation is conserved). These concepts are not sufficient to determine a unique solution for the wing and wake circulation distribution. Then, the Kutta condition (smooth flow off the sharp trailing edge) comes to play an essential role in the problem closure. That is, no flow around the sharp edge and hence, even within the framework of potential flow, the velocity has to be finite at the edge. Finally, in order to obtain an explicit analytical solution of the governing equation (the Laplace’s equation in the velocity potential in this case), one more assumption is usually adopted: Assuming small disturbance to the mean flow (i.e., the vorticity sheet convects by the mean flow velocity: flat wake assumption) completes the framework of the classical theory. In summary, the classical theory of unsteady aerodynamics is based on replacing the airfoil and the wake by vorticity distributions (singularities) that satisfy the Laplace’s equation everywhere in the flow field except at the surface of singularities. Three main conditions are applied: (1) no-penetration boundary condition (fluid velocity is parallel to the wing surface), (2) The Kutta condition (smooth flow off the sharp trailing edge), and (3) the conservation ∗ Corresponding author. E-mail address: [email protected] (H.E. Taha). https://doi.org/10.1016/j.jfluidstructs.2020.102868 0889-9746/© 2020 Elsevier Ltd. All rights reserved.

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H.E. Taha and A.S. Rezaei / Journal of Fluids and Structures 93 (2020) 102868

Nomenclature b C (k) CP CL G(s) GL (s), GΓ (s) (m) Hn j k Kn kdc , khf LC , LNC ma Py R s, s˜ t U u

w x y

α δ ϵ φ Γ , Γ˜ Γ0 γ κ L

ν ω ρ τ χ

Half chord Theodorsen function (lift deficiency function or lift frequency response function) Pressure coefficient Lift coefficient Transfer function of the linear dynamical system in Laplace domain Lift and circulation transfer functions Hankel function of mth kind of order n √ −1 Reduced frequency modified Bessel function of second kind of order n Dc gain and high frequency gain Circulatory and non-circulatory lift forces Added mass Momentum in the y-direction normal to the airfoil Reynolds number Laplace variable and non-dimensional Laplace variable Time variable Free stream velocity Control inputs for a dynamical system airfoil velocity normal to the surface (positive downward) Coordinate along the airfoil chord Output of a dynamical system Angle of attack or pitching displacement Thickness of the Stokes layer Trailing edge angle Wagner function Circulation and non-dimensional circulation Quasi-steady circulation Circulation distribution per unit length κ = 2 − ϵ/π Laplace operator Kinematic viscosity in m2 /s Angular frequency of the harmonic motion in rad/s Air density Non-dimensional time Vector of state variables of a dynamical system

of total circulation. This formulation along with the flat wake assumption constitute the classical theory of unsteady aerodynamics. The above formulation of the classical theory of unsteady aerodynamics was extensively used throughout the years. In 1925, Wagner (0000) used this formulation to solve the indicial problem (lift response due to a step change in the angle of attack). In 1935, Theodorsen (1935) used the same formulation to solve the frequency response problem (steady state lift response due to harmonic oscillation in the angle of attack). In 1938, Von Karman and Sears (1938) provided a more general and elaborate representation of the classical formulation. Also, the works of Küssner (1929) for the sharp edged gust problem, Schwarz (1940) for the frequency response problem, and Loewy (1957) for the returning wake problem worth mentioning. It should be noted that while there may be different approaches within this framework, these efforts are essentially equivalent. For example, the lift frequency response functions derived by Von Karman and Sears (1938) and Schwarz (1940) are identical to the Theodorsen’s function (Theodorsen, 1935), in spite of the three approaches being different. In addition, Garrick (1938) showed that the Theodorsen function and the Wagner function form a Fourier transform pair. One of the well-known outcomes from Theodorsen’s seminal effort (Theodorsen, 1935) is the classification of lift into circulatory and non-circulatory components. The latter is due to the instantaneous acceleration of the airfoil. It represents the force required to accelerate the fluid surrounding the airfoil. Therefore, it is as if one has to accelerate not only the mass

H.E. Taha and A.S. Rezaei / Journal of Fluids and Structures 93 (2020) 102868

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Fig. 1. Illustration of the steady (quasi-steady) and unsteady behavior of lift due to a step change in the angle of attack α .

of the airfoil, but also the mass of the surrounding fluid; hence, it is given the name added mass or virtual mass. This ‘‘lift’’ force, which is more of a resistance force, is characterized as non-circulatory because the associated flow field has no net circulation around the airfoil. Therefore, this non-circulatory lift force exists only in an unsteady flow; in steady flows, lift is solely due to circulation according to the Kutta–Joukowsky lift theorem (Schlichting and Truckenbrodt, 1979). It should be noted that, according to Theodorsen’s model (Theodorsen, 1935), the non-circulatory flow satisfies the no-penetration boundary condition but does not satisfy the Kutta condition; i.e., the velocity is not finite at the trailing edge. As such, since Theodorsen formulated the problem in the cylinder domain (relying on the conformal mapping between a cylinder and a flat plate), he added vortices in the wake that, together with their image vortices inside the cylinder, (i) do not disturb the no-penetration boundary condition (i.e., maintain the cylinder as a stream line), (ii) satisfy the conservation of circulation, and (iii) together with the non-circulatory flow satisfy the Kutta condition at the trailing edge. This last condition was used to write the governing integral equation for wake circulation. The other lift component (circulatory component) in Theodorsen’s formulation is the lift contribution from the second flow field that was added to satisfy the Kutta condition. Its name and Theodorsen’s formulation give the impression that the circulatory lift is due to circulation. However, we show that this is not accurate; the circulatory lift is not only due to circulation and its rate of change. In this paper, we study the dynamics of unsteady lift and circulation from a system dynamics perspective. In particular, we focus on the high-frequency gain of lift and circulation transfer functions and their relations to the initial values of lift and circulation immediately after an impulsive start. Using the unsteady thin airfoil theory and Peters inflow model (Peters et al., 1995; Peters, 2008), we show that the circulatory lift (as defined by the pioneers Wagner (0000) and Theodorsen (1935)) is not solely due to circulation and its rate of change in an unsteady flow; it has some contribution from the airfoil’s acceleration (i.e., it enjoys some added-mass aspects). In fact, we show that such a classification (circulatory/noncirculatory) is arbitrary. Moreover, we show that a non-circulatory motion (a motion that is expected to generate no circulatory lift) may induce circulation over the airfoil. This latter observation is demonstrated via laminar numerical simulation of the Navier–Stokes equations on a dynamic mesh around NACA 0012 undergoing pitching motion in a quiescent fluid (no free stream). The theme and illustrations will be imbued with the dynamical-systems aspect of unsteady aerodynamics. 2. Dynamical systems background One of the main characteristics that distinguishes unsteady aerodynamics from steady aerodynamics is the dynamical behavior of the former. For example, it is well-known that the steady lift coefficient of a flat plate at a small angle of attack is given by CL = 2π α . If the angle of attack is time-varying, this relation is written as CL (t) = 2πα (t), which implies an instantaneous lift development. That is, the angle of attack α (t) at the instant t dictates a lift force at the same instant, which is not physically possible. There must be a finite time (i.e., lag) for the lift to build up, however small this time may be. This type of instantaneous behavior is called quasi-steady which follows the same time-variation of the angle of attack (more specifically, the angle of attack at the three-quarter chord point Leishman, 2006; Bisplinghoff et al., 1996), as shown in Fig. 1 for a 5◦ step change in the angle of attack. In reality, the lift will take some time to build up and reach the steady value corresponding to the given angle of attack, as shown in the schematic of Fig. 1. From the above discussion, one can infer that the angle of attack (or airfoil motion) is the input to the aerodynamic system and the lift is one of its main outputs. For an unsteady aerodynamic model, the CL -α relation cannot be algebraic (e.g., CL (t) = f (α (t)) for any function f ). It also cannot be in the form CL (t) = f (α (t), α˙ (t), α¨ (t)). Rather, it has to be a

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H.E. Taha and A.S. Rezaei / Journal of Fluids and Structures 93 (2020) 102868

dynamical relation described by a differential equation for CL (e.g., dtL = f (α (t), α˙ (t), α¨ (t))), transfer function αL , state space model, frequency response, etc. In other words, the CL -α relation is no longer a simple gain (2π ), but rather a dynamical system. It should be noted that while this view is quite natural and intuitive, it was not explicitly described by the efforts of the early pioneers (Wagner, 0000; Theodorsen, 1935; Von Karman and Sears, 1938), obviously because the dynamical-systems theory was not mature in their period as it currently is. Some of the efforts that adopted this dynamical-systems perspective includes Beddoes (1983, 1984), Leishman and Nguyen (1990), Peters et al. (1995), Hemati et al. (2016), Brunton et al. (2014), Zakaria et al. (2015, 2017), Izraelevitz et al. (2017), and Shehata et al. (2018). Having the above description in mind, we believe that it would be better to provide a quick review of some of the standard dynamical-systems tools that will be used throughout the paper to describe the unsteady lift and circulation dynamics. There are two main mathematical representations of linear dynamical systems (Ogata and Yang, 0000): the classical representation via transfer function and the modern representation after Kalman (1963) in a state space form. This latter representation is written as dC

χ˙ (t) y(t)

C

[A]χ(t) + [B]u(t), [C ]χ(t) + [D]u(t),

= =

where χ is a vector of state variables (that define the system), u is the input vector, y is the output vector, and the matrices A, B, C , D are constant matrices that define the system. On the other hand, the transfer function is defined as the Laplace transform of the output divided by the Laplace transform of the input at zero initial conditions (Ogata and Yang, 0000) G(s) =

⏐ Y (s) ⏐ = , ⏐ L{u(t)} y(0)=0,˙y(0)=0,... U(s) L{y(t)} ⏐

where L is the Laplace transform and s is its variable. Also, there are two main response functions that define the characteristics of a linear dynamical system: the step response (transient response due to a step input) and the frequency response (the steady state response due to a harmonically oscillating input). It should be noted that any of the two mathematical representations or any of the two response functions completely define the dynamical characteristics of a linear system. Given any of the four, one can determine the other three. For example, given a state space model (A, B, C , and D), the transfer function can be obtained as G(s) = C [sI − A]−1 B + D, from which the step response can be determined as ys (t) = L−1 {G(s). 1s } and the frequency response can be determined by substituting s = jω (Ogata and Yang, 0000). Table 1 shows an illustration for the step and frequency response functions for the following transfer function G(s) =

0.3s2 + s + 3 s2

⇒ G(s = jω) =

+s+2

3 − 0.3ω2 + jω 2 − ω2 + jω

whose state space matrices can be written as

⎛ ⎞ ( ) x˙ 1 x˙ 2 y

[

0 −2

=

=

2.4

[

]

1 ⎜x1 ⎟ + −1 ⎝ ⎠ x2

( ) ] x1

0.7

x2

( )

0 u, 1

+ 0.3u.

The focus of this review is to recall the two concepts of the dc gain kdc and the high-frequency gain khf . The dc gain simply represents the steady amplification: it is the steady state value of the output due to a unit step input (kdc = limt →∞ y(t)). From the step response in the time domain, shown in Table 1, one can deduce its value of 1.5. It can be determined from the transfer function as kdc = lims→0 G(s), which results in 1.5 when substituting s = 0 in the G(s) above. It can also be determined from the frequency response in Table 1: the magnitude at zero frequency which gives the same value of 1.5. On the other hand, the high-frequency gain represents the other end of the spectrum (at infinite frequency and zero time). That is, it is defined as khf = limω→∞ |G(jω)|: the infinite-frequency amplification, whose value is found to be 0.3 from the frequency response in Table 1. It can also be determined from the transfer function as kdc = lims→∞ G(s). As such, its value is zero for a strictly proper transfer function (degree of numerator is strictly lower than the degree of denominator). If the two polynomials have the same degree, then khf is simply equal to the ratio of the coefficients of the highest power. This ratio is also the value of D of the corresponding state space model. From the definition of khf (the gain at infinite-frequency) and D, it is understood that a system with nonzero khf = D will have some instantaneous response. In particular, khf (or D) represents the instantaneous initial response in the time domain due to a unit step input, as can be seen from the step response in Table 1: ys (0) = khf = 0.3. In this case of non-zero khf , the high-frequency phase is either 0 or 180◦ . In general, the high-frequency limit of the phase angle is −90◦ multiple of the relative degree between the numerator and denominator provided that the system is minimum phase (no zeros in the right half of the complex plane).

H.E. Taha and A.S. Rezaei / Journal of Fluids and Structures 93 (2020) 102868

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Table 1 The two main response functions describing a dynamical system. Step response

Frequency response

u(t) = 1 for all t ≥ 0 ys (t) = L−1 {G(s). 1s }

u(t) = A cos ωt yss (t) = A|G(jω)| cos (ωt + ̸ G(jω))

2

In conclusion, the numerator’s degree of a transfer function cannot be higher than the denominator’s degree for causality (i.e., the current output must depend on the current and past inputs). If there is dependence on the current input (i.e., non-zero D or khf ), then it implies that there is some immediate response from the system, which is rare to find in mechanical systems. The most common case is for a strictly proper transfer function where khf = D = 0 for which the input u(t) at an instant t cannot dictate the output y at the same instant or even part of it. Rather, it may dictate its rate of change at this instant, affecting its value in the future. 3. Unsteady lift dynamics: Step response, frequency response, and transfer function In the previous section, we reviewed the two main response functions describing a linear dynamical system: the step response and the frequency response. For the aerodynamic system whose input is the airfoil motion (e.g., angle of attack) and output is the unsteady lift, these two fundamental response functions have been determined in the first half of the past century, assuming potential flow. Wagner (0000) solved the indicial problem (lift response due to a step change in the angle of attack) and Theodorsen (1935) solved the frequency response problem (steady state lift response due to harmonic oscillation in the angle of attack). Therefore, if the lift transfer function is denoted by GL (s), then the Wagner function φ (t) will be given by φ (t) = L−1 {GL (s). 1s } and the Theodorsen function C (ω) will be given by C (ω) = GL (s = jω). It should be noted that both Wagner and Theodorsen have non-dimensionalized their solutions by the free-stream velocity U and the semi-chord length b. That is, the Wagner function is given in terms of the non-dimensional time τ = Ut (traveled distance b in semi-chord lengths) and the Theodorsen function is given explicitly in terms of the non-dimensional frequency k = ωUb , called the reduced frequency, as (2)

C (k) =

H1 (k) (2) H1 (k)

+

(2) iH0 (k)

=

K1 (jk) K1 (jk) + K0 (jk)

,

(1)

(m)

where Hn is the Hankel function of mth kind of order n and Kn is the modified Bessel function of second kind of order n. Consequently, taking the Laplace transform from the non-dimensional time (τ -domain), one reaches the non-dimensional Laplace (s˜-domain) and the corresponding frequency domain is in terms of the reduced frequency k. In other words, φ (τ ) = L−1 {GL (s˜). 1s˜ } and C (k) = GL (s˜ = jk). Thus, from Eq. (1), one can write the lift transfer function in the Laplace domain as GL (s˜) =

K1 (s˜) K1 (s˜) + K0 (s˜)

.

(2)

Unlike Theodorsen function, the Wagner function φ (τ ) is not written explicitly in τ , but rather is given by the integral (Wagner, 0000; Von Karman and Sears, 1938)

φ (τ ) =

τ

∫ 0

µ(σ ) √

1+τ −σ (1 + τ − σ )2 − 1

dσ ,

(3)

where the function µ is governed by the integral equation τ

∫ 0

√ µ(σ )

2+τ −σ

τ −σ

dσ = 1.

(4)

Wagner (0000) provided tables for the functions µ and φ in terms of τ . As discussed above, either φ (τ ) or C (k) completely characterize the inviscid lift dynamics. For example, the circulatory lift response due to an arbitrary time-varying angle of attack (input) can be determined from the step response φ (τ ) using

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Fig. 2. The step and frequency response functions (Wagner and Theodorsen functions) of the inviscid circulatory lift.

the Duhamel superposition principle (Garrick, 1938; Beddoes, 1983, 1984; Leishman and Nguyen, 1990; Bisplinghoff et al., 1996; Leishman, 2006; Taha et al., 2013, 2014)

[

LC (τ ) = 2π ρ U 2 b α (0)φ (τ ) +

τ

∫ 0

dα (σ ) dσ

] φ (τ − σ )dσ ,

(5)

where ρ is the fluid density. On the other hand, the same response due to an arbitrary input can be determined from the frequency response C (k) using the Fourier transform (Garrick, 1938; Bisplinghoff et al., 1996) LC (τ ) = ρ U 2 b

∫

∞

αˆ (k)C (k)ejkτ dk,

(6)

−∞

where αˆ (k) = −∞ α (τ )e−jkτ dτ is the Fourier transform of the angle of attack aerodynamic input. Using this analysis, and realizing that the Fourier transform of a unit step angle of attack is αˆ (k) = jk1 , Garrick (1938) showed that the Wagner function and the Theodorsen function form a Fourier Transform pair:

∫∞

φ (τ ) =

1 2π

∫

∞ −∞

C (k) jk

jkτ

e

∫

∞

dk and C (k) = jk

φ (τ )e−jkτ dτ .

(7)

−∞

Fig. 2 shows the inviscid circulatory lift step and frequency responses (i.e., Wagner and Theodorsen functions). While 2 Theodorsen function is shown exactly according to Eq. (1), Fig. 2(a) shows Garrick’s approximation φ (τ ) ≃ ττ + of the +4 Wagner function. Recalling the review provided in Section 2, one can clearly see that the high-frequency gain associated with the inviscid circulatory lift is half: khf = φ (τ = 0) = lim |C (k)| = lim GL (s˜) = k→∞

s˜→∞

1 2

.

Actually, any approximation of the Wagner function in the time domain will have to satisfy this property; e.g., the Garrick’s +2 approximation ττ + (Garrick, 1938) or the two-state approximations of Jones (1938, 1945) 4

φ (τ ) ≃ 1 − A1 e−b1 τ − A2 e−b2 τ . Therefore, A1 + A2 = 21 in all of their approximations. On the other side, any finite-state approximation of the Theodorsen function in the frequency domain or its transfer function in the Laplace domain will also have to satisfy this property; 2 e.g., the Pade approximations of Theodorsen function by Vepa (0000) 2s˜s˜++00..55 , 2s˜s˜2++12..55s˜s˜++00..375 , and the several ones by 375 Dinyavari and Friedmann (1984). This fact implies that half of the steady lift is immediately generated (instantaneously developed) without any lag; the system dynamics apply to the other half. This behavior is not physically appealing since there must be a finite time for the lift to develop, particularly because we are concerned with the circulatory lift (it is understandable that the non-circulatory lift responds instantaneously based on the instantaneous acceleration, at least in the absence of viscosity and compressibility). However, this behavior cannot be true for the lift due to circulation even when assuming potential flow because circulation cannot develop instantaneously, as explained in the next section.

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Fig. 3. Initial flow over an airfoil right after an impulsive start. The flow looks like an inviscid flow with zero circulation.

4. The dynamics of circulation development 4.1. Fluid mechanics perspective It is well-known that at the very first moment (t = 0+ ) right after an airfoil impulsive start, the flow looks like an inviscid flow without circulation, as shown in Fig. 3 and discussed in Prandtl’s lectures (Tietjens and Prandtl, 1934, pp. 158–168; Goldstein, 1938, pp. 26–41; Milne-Thomson, 1996, pp. 91–92; and Schlichting and Truckenbrodt, 1979, pp. 33–35). That is, the initial response (high-frequency response/gain) of the bound circulation around the airfoil is zero. This behavior can be physically explained as follows. For an impulsive starting airfoil, at t = 0+ , the velocity everywhere is zero except on the airfoil. There is an infinitely thin boundary layer taking care of the no-slip, but at t = 0+ , it has to be infinitely thin so that if the airfoil is enclosed with any contour, no matter how close it gets to the airfoil, the contour will pass through points outside of this layer. On this contour, the diffusion term in the Navier–Stokes equation is zero (i.e., the flow behaves like an inviscid flow temporarily); as such, if this contour is allowed to expand with the fluid (to contain the same fluid particles), the Kelvin’s circulation theorem will imply that the circulation remains constant on this contour. Hence, if the contour circulation ΓC was zero before the impulsive start, it would remain constant throughout the entire excursion. In particular, at t = 0+ , the contour did not have a chance to expand beyond the airfoil boundary and therefore, the contour circulation ΓC matches the bound circulation Γ around the airfoil implying that limt →0+ Γ (t) = 0 (i.e., no jump in circulation development), which is physically intuitive because circulation needs time to build up even in potential flow. At this instant, the starting vortex is forming, but it has not shed yet and as such, it is part of the airfoil (ideally at the trailing edge). It should be noted that although the two circulations match at the initial moment (ΓC (t = 0+ ) = Γ (t = 0+ ) = 0), their rates of change do not match: Of course, Γ˙ C ≡ 0 throughout the excursion because of Kelvin’s theorem, while Γ˙ ̸ = 0. In fact, the initial development of a point vortex (or vortex sheet) from a sharp edge has been extensively studied in literature (Rott, 1956; Pullin, 1978; Chow and Huang, 1982; Graham, 1983; Cortelezzi and Leonard, 1993; Jones, 2003; Pullin and Wang, 2004; Li and Wu, 2015). The common technique for studying the evolution of a vortex sheet has been the Rott–Birkhoff equation simultaneously with the Kutta condition at the edge (Pullin, 1978; Jones, 2003; Pullin and Wang, 2004). The Rott–Birkhoff equation describes the evolution of the curve representing a planar vortex sheet in twodimensional, inviscid, incompressible flow. The tangential velocity across the sheet is discontinuous and the flow outside the sheet is irrotational. Graham (1983) showed that for a starting flow, the dynamics of a vortex sheet can be wellapproximated by a point vortex model. In this case, the Brown–Michael model (governing the dynamics of a point vortex of time-varying strength (Brown, 1954; Hussein et al., 2018)) simultaneously with the Kutta condition has been the most common technique (Rott, 1956; Graham, 1983; Cortelezzi and Leonard, 1993; Michelin and Smith, 2009). The results of these efforts can be summarized as follows. If the airfoil has a constant angle of attack α in a starting ¯ m , Graham (1983) showed that the vortex strength will grow as Γ ∼ t (2κ m+1)/(2κ−1) , flow with a speed profile U = Ut where κ is related to the trailing edge angle ϵ as κ = 2 − ϵ/π . This result reduces to Γ ∼ t (4m+1)/3 in the case of a flat plate (ϵ = 0 or κ = 2) or an airfoil with a cusped trailing edge (e.g., Joukowsky airfoil), which is conforming with the results of Jones (2003) and Pullin and Wang (2004) for a flat plate with two vortex sheets emanating from the leading and trailing edges; and the results of Cortelezzi and Leonard (1993) for a point vortex shed from the sharp edge of a semi-infinite plate. This result further reduces to Γ ∼ t 1/3 in the case of an impulsively starting flow (m = 0), which is conforming with the results of Pullin for a vortex sheet (Pullin, 1978). In all of these cases, the initial circulation is zero: Γ (t = 0) = 0, supporting the physical description presented above. 4.2. Dynamical systems perspective: Frequency response and transfer function It is interesting to point out that the frequency response of circulation is not given by the Theodorsen function C (k). Following Schwarz (1940), one can write the frequency response of circulation as (Schwarz, 1940; Bisplinghoff et al., 1996)

Γ −2e−jk ( ), (k) = (2) (2) Γ0 jkπ H1 (k) + jH0 (k)

(8)

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H.E. Taha and A.S. Rezaei / Journal of Fluids and Structures 93 (2020) 102868

Fig. 4. Comparison between the frequency response functions of the circulatory lift and circulation in potential flow.

where Γ0 is the quasi-steady circulation. Fig. 4 shows a comparison between the lift frequency response C (k) and the circulation frequency response as given by Eq. (8). The potential-flow lift frequency response (Theodorsen function C (k)) does not represent a physical system in mechanics: from a dynamical system perspective, it does not act as a low pass filter. This characteristic is apparent from its high frequency limit limk→∞ C (k) = 12 : a non-zero high-frequency gain with zero phase lag. That is, there exists some component in the circulatory lift that responds instantaneously; i.e., it depends algebraically on the current input (airfoil motion) without dynamics (lag). This characteristic is apparent in the non-zero initial-time response due to a step input (φ (τ = 0) = 21 ). On the other hand, the high-frequency limit of the circulation transfer function is given by

⏐ ⏐ ⏐Γ ⏐ Γ ⏐ lim (k)⏐⏐ = 0 and lim ̸ (k) = −π/4, k→∞ ⏐ Γ0 k→∞ Γ0 which points to a strictly proper transfer function. That is, in contrast to the circulatory lift response, there is no instantaneous component of circulation. Indeed, the circulation development possesses lag similar to a mechanical system with a strictly proper transfer function even within the framework of potential flow. Therefore, limt →0 Γ (t) = 0 in an impulsive start, which conforms with the flow physics discussed above. 5. Relation between circulatory lift and circulation 5.1. Dynamical systems perspective We note that Eq. (8) can be re-written as

−2 e−jk Γ (k) = C (k). Γ0 π jk H1(2) (k)

(9)

As such, realizing that K1 (jk) = − π2 H1 (k), we write the following relation between the circulatory lift transfer function GL (s˜) (i.e., corresponding to Theodorsen function: GL (ik) ≡ C (k)) and the circulation transfer function, denoted by GΓ (s˜) (2)

GΓ (s˜) =

1 e−˜s s˜ K1 (s˜)

GL (s˜),

(10)

Eq. (10) is quite revealing for the relation between the dynamics of the circulatory lift and circulation in potential flow. Clearly, there is a time-integrator 1s˜ and a time-delay e−˜s between the circulatory lift development and circulation development. However, the term K 1(s˜) acts as a differentiator in the low frequency range (K1 (z) ∼ 1z as z → 0 Wolfram, 1 1998, pp. 4). Therefore, it cancels the effect of the integrator, hence, the circulation development follows the lift evolution with a time-delay in the low-frequency range. On the other hand, the term K 1(s˜) cancels the delay effect in the high frequency range (K1 (z) ∼

√π

2z

1

e−z as z → ∞ Press et al., 1992, pp. 236) leading to a circulation transfer function

H.E. Taha and A.S. Rezaei / Journal of Fluids and Structures 93 (2020) 102868

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Fig. 5. Relation between the airfoil motion, circulatory lift, and circulation development in potential flow.

Fig. 6. Comparison between the frequency response functions of the circulatory lift and circulation in potential flow along with the approximation of Lomax et al. (1952) and the approximation given in Eq. (11).

of the form √1 which explains the zero-high frequency gain and the −45◦ high-frequency phase. Fig. 5 shows the s˜ relation between the airfoil motion (quasi-steady behavior), circulatory lift, and circulation development in potential flow. It is found that LC leads Γ in potential-flow unsteady aerodynamics; there is a delay from lift to circulation in the low-frequency range and a fractional integrator √1 in the high-frequency range. s˜

From the above discussion, one can write the relation between the circulatory lift and circulation in the low-frequency range as GL (s˜) ≃ GΓ (s˜)es˜ ≃ GΓ (s˜) 1 + s˜ ,

(

)

(11)

which yields

φ (τ ) ≃ Γ˜ (τ ) + Γ˙˜ (τ ), similar to the approximation by Lomax et al. (1952) of the circulatory lift in terms of circulation and its rate of change: Γ φ = Γ˜ + 1.5Γ˙˜ , where Γ˜ = Ub is the non-dimensional bound circulation. Fig. 6 shows that the approximation (11) is more accurate in capturing the circulatory lift dynamics than that of Lomax et al. (1952), and that both approximations do a better job capturing the magnitude than the phase. Moreover, it is clear that these approximations are not valid in the high-frequency range, which is the main focus of this paper. In other words, the notion that the circulatory lift is due to circulation and its rate of change may be valid in the small frequency range, which is intuitively expected since in the limit to zero frequency (i.e., steady), lift is solely due to circulation (Izraelevitz et al., 2017). However, in the high-frequency range, there are other contributions (acceleration contributions) that violate this notion, as explained below using the thin airfoil theory. On the side, it is interesting to recall the efforts of Graham (1983) and Chow and Huang (1982) regarding the initial ¯ m . They showed that the lift force is given by two contributions: one lift in a starting flow with a speed profile U = Ut proportional to dU (due to the airfoil acceleration, hence, classified as non-circulatory or added-mass) and another one dt proportional to U 2 (due to the airfoil velocity). Following consensus in literature, they classified the latter component as circulatory. Graham showed that this circulatory component LC grows as t (4m+3−2κ )/(2κ−1) , which reduces to t (3−2κ )/(2κ−1) for an impulsively starting flow (m = 0), in contrast to the result LC ∼ t (2−κ )/(κ−1) by Chow and Huang (1982). Graham’s result using a point vortex model points to a lift singularity at t = 0 for any trailing edge angle, while the results of Chow and Huang (assuming an initially flat wake element) imply a finite initial lift that matches Wagner’s behavior (Wagner, 0000) (i.e., half of the steady lift) for a cusped trailing edge and a zero initial lift for any finite trailing edge angle. Note

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that the flat wake assumption is very common in the early efforts by the pioneers Wagner (0000), Theodorsen (1935), Von Karman and Sears (1938) among others. Graham showed that the flat wake assumption does not accurately represent the physics of a rolling up vortex sheet at any time in a starting flow, however short, for any exponent m and any trailing edge angle. Rather, a point vortex model provides a satisfactory approximation of the sheet roll up dynamics. In contrast, he showed that the initially flat wake element assumption may be valid in the case of an airfoil gradually changing its angle of attack in a steady stream. These facts imply that the flow dynamics and lift response due to a step change in the angle of attack may be different from those associated with an impulsively starting flow at a fixed angle of attack (i.e., step change in the airfoil speed or airstream); Wagner’s flat wake assumption is closer to represent the former problem (step in α ) than the latter problem. In either case, the circulation initial response is zero while the initial response of the circulatory lift is either finite or singular. All of the above evidences support the fact that the circulatory lift is not the lift due to circulation (nor its rate of change). The pioneers (Theodorsen, 1935; Wagner, 0000; Von Karman and Sears, 1938) opted to define the circulatory lift in a different way. This point becomes clearer when the problem is formulated using thin airfoil theory, which is explained in the next section. 5.2. Thin airfoil theory perspective To show how the steady lift is solely due to circulation (L = ρ U Γ ) whereas the unsteady ‘‘circulatory’’ lift is not lift due to circulation (nor its rate of change), we recall the thin airfoil theory. In such a theory, a thin airfoil is represented by a circulation distribution along the chord line (x-axis), as shown in Table 2, and the no-penetration boundary condition results in the integral equation for circulation 1

∫

2π

b

−b

γ (ζ )

dζ = w (x), ∀ x ∈ [−b, b],

x−ζ

(12)

∫b

where γ is the circulation distribution over the airfoil (i.e., Γ = −b γ (x)dx) and w is the airfoil velocity normal to the surface (positive downward). This integral equation is typically solved assuming the series solution

[ γ (ϕ ) = 2U A0 tan ϕ/2 +

∞ ∑

] An sin nϕ ,

(13)

n=1

where x = b cos ϕ . Then, the no-penetration boundary condition (12) implies that the An ’s are the coefficients of the Fourier cosine representation of the normal velocity; that is, A0 =

1

π

∫

π

w(ϕ )dϕ, and An =

0

2

π

π

∫

w(ϕ ) cos nϕ dϕ, ∀ n ≥ 1.

0

After solving for the series coefficients, and consequently the circulation distribution γ , the normalized pressure difference can be determined from the steady Bernoulli equation as 2

γ (x), (14) U which implies that the local pressure is dependent only on the local circulation distribution; that is, ∆CP (x) and γ (x) possess the same distribution over the airfoil resulting in ∆CP (x) =

Γ˜ = CL = 2π (A0 + A1 /2) ,

(15)

which is equivalent to L = ρ U Γ . The unsteady thin airfoil theory has been introduced by Katz and Plotkin (2001), Peters et al. (1995), Peters (2008), and Ramesh et al. (2013) among others. In contrast to the steady thin airfoil theory, the wake effects must be taken into account for an unsteady analysis, as shown in Table 2. That is, in addition to the circulation distribution γ bound the airfoil, there is a sheet of vorticity in the wake whose circulation distribution γw is also unknown. As such, the no-penetration boundary condition (12) becomes 1 2π

∫

b

−b

γ (ζ , t) 1 dζ + x−ζ 2π

∞

∫ b

γw (ζ , t) dζ = w (x, t), ∀ x ∈ [−b, b], x−ζ

(16)

Assuming a similar series solution to that of the steady case (13), but with time-varying coefficients

[ γ (ϕ, t) = 2U A0 (t) tan ϕ/2 +

∞ ∑

] An (t) sin nϕ

n=1

and expanding the normal velocity w in a Fourier cosine series similar to the steady case

[ w(x, t) = U B0 (t) +

∞ ∑ n=1

] Bn (t) cos nϕ ,

(17)

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Table 2 Summary of the formulations of the steady and unsteady thin airfoil theory. Steady

Unsteady

= w(x) ] ∑ γ (ϕ ) = 2U A0 tan ϕ/2 + ∞ n=1 An sin nϕ [ ] ∑∞ w(ϕ, t) = U B0 + n=1 Bn cos nϕ An = Bn ∆CP (x) = U2 γ (x) Γ˜ = 2π (A0 + A1 /2) = 2π (B0 + B1 /2)

γw (ζ ,t) dζ = w (x, t) x−ζ ] ∑ γ (ϕ, t) = 2U A0 (t) tan ϕ/2 + ∞ n=1 An (t) sin nϕ [ ] ∑∞ w(ϕ, t) = U B0 (t) + n=1 Bn (t) cos nϕ

CL = 2π (B0 + B1 /2)

CL = 2π ⎢B0 + B1 /2 − λ0 +

1 2π

γ (ζ ) dζ −b x−ζ [

∫b

1 2π

γ (ζ ,t) dζ −b x−ζ [

∫b

1 2π

∫∞ b

An + λn = Bn ∆CP (x, t) = U2 γ (x, t) +

∫x 2 ∂ γ (ζ , t)dζ U 2 ∂ t −b ˜ Γ = 2π (A0 + A1 /2) = 2π (B0 + B1 /2 − λ0 − λ1 /2) ⎡ ⎤ ⎢ ⎢ ⎣

L = ρU Γ

+



Circulatory



1(

2

)⎥ ⎥ ⎦

B˙ 0 − B˙ 2 /2 ⎥



Non−Circulatory

LC = ρ U Γ + πρ U bλ1 2

the no-penetration boundary condition (16) implies An + λn = Bn ∀ n ≥ 0,

(18)

where λn ’s are Peters inflow coefficients (Peters et al., 1995; Peters, 2008): the coefficients of the Fourier cosine representation of the downwash λ(x, t) on the plate due to wake vorticity. That is, λ is defined as

λ(x, t) =

1 2π

∞

∫ b

[ ] ∞ ∑ γw (ζ , t) dζ = U λ0 (t) + λn (t) cos nϕ . x−ζ n=1

Unlike the steady case where the no-penetration boundary condition was sufficient to determine γ (or its coefficients An ’s), Eq. (18) has two unknowns An , λn for each n and, hence, is not sufficient. To achieve closure in the unsteady case, a relation must be established between the wake vorticity and bound circulation. Peters et al. (1995) and Peters (2008) utilized the fact that the wake convects freely Dγw Dt

=U

∂γw (x, t) ∂γw (x, t) + =0 ∂x ∂t

along with the Kutta condition (or conservation of total circulation) to derive the following equation U

1 Γ˙ ∂λ(x, t) ∂λ(x, t) + = , ∂x ∂t 2π b − x

(19)

which results in linear differential equations governing the inflow coefficients λn ’s (i.e., a state space model for λn ’s): Peters finite state model. Eqs. (18), (19) can be used to determine the coefficients An ’s and λn ’s after truncating the series representations at a certain n. Peters et al. (1995) and Peters (2008) showed that eight terms (states) may be sufficient to capture the inviscid unsteady lift dynamics. Having solved for the circulation distribution γ (and its coefficients), the non-dimensional circulation will have the same form

Γ˜ = 2π (A0 + A1 /2) = 2π (B0 + B1 /2 − λ0 − λ1 /2)

(20)

as in the steady case (15) because γ has the same series representation. However, the unsteady Bernoulli equation implies

∆CP (x) =

2 U

γ (x) +

2 ∂ U2

∂t

∫

x

γ (ζ )dζ ,

(21)

−b

which is different from its steady counterpart (14); the second term is due to the time-derivative term in the unsteady Bernoulli equation. Consequently, in contrast to the steady case, Eq. (21) implies that the local pressure distribution may be different from the circulation distribution. Integrating the pressure distribution over the airfoil and utilizing Eq. (19),

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H.E. Taha and A.S. Rezaei / Journal of Fluids and Structures 93 (2020) 102868

one obtains

⎡

⎤

⎢ )⎥ 1( ⎥ ˙ ˙ CL = 2π ⎢ ⎣B 0 + B1/2 − λ0 + 2 B0 − B2 /2 ⎦ .    Circulatory

(22)

Non−Circulatory

The common classification of the total unsteady lift into circulatory and non-circulatory components can be clearly seen from Eq. (22). First, it should be pointed out that such a classification is arbitrary. For example, one can add and subtract λ1 /2 to the terms inside the square bracket in Eq. (22) and collect the terms that produce Γ˜ referring to them as circulatory lift (lift due to circulation in this case) and the rest are non-circulatory terms. This classification has been adopted by Fung (1969). In contrast, the most common classification, adopted by Wagner (0000), Theodorsen (1935), Von Karman and Sears (1938), among many others Bisplinghoff et al. (1996) is that the last two terms (acceleration terms) in Eq. (22) represent the non-circulatory loads and the first three terms (velocity terms) represent the circulatory loads. According to this common classification, the circulatory lift is not lift due to circulation (i.e., LC ̸ = ρ U Γ ). Also, it is not lift due to circulation and its rate of change. Rather, they are related as LC = ρ U Γ + π ρ U 2 b λ 1 .

(23)

Moreover, Eq. (19) can be used to write λ1 as

λ1 =

b

[

U

] Γ˙˜ ˙ ˙ − λ0 + λ2 /2 , π

(24)

which implies that λ1 is related to not only the rate of change of circulation, but also acceleration terms. That is, the common definition of circulatory lift is not only lift due to instantaneous circulation and its rate of change. Eqs. (23), (24) imply that the circulatory lift LC is obtained from the lift due to circulation ρ U Γ after adding some acceleration terms, which indicates that the circulatory lift (with its common definition) may have some ‘‘added mass’’ effects. This conjecture may explain the reason behind the non-zero high-frequency gain of the circulatory lift frequency response (φ (0) = limk→∞ |C (k)| = lims˜→∞ GL (s˜) = 12 ): half of the steady-state lift is immediately generated. This instantaneous component of LC must be the added mass part of its definition. This fact may be supported by recalling the efforts of Chow and Huang (1982). They recovered Wagner’s initial lift φ (0) from purely added mass considerations. Assuming an initially flat wake element, they accounted for the initial lift from the added-mass effect on the elongated airfoil (airfoil plus the wake element) (Jones, 1940). That is, in a short time interval ∆t right after an impulsive start of an airfoil at a small angle of attack α , a flat wake element of length U ∆t is shed from the airfoil trailing edge (along the chord direction); therefore, the airfoil chord 2b will be elongated to 2b + U ∆t, resulting in an added mass of ma = π ρ (2b + U ∆t )2 /4 for the combined airfoil-wake system, which in turn results in y-momentum of Py = −ma U α . Therefore, the early lift is given by dPy

˙ a, = ma U˙ α + U α m dt where the first term is the acceleration effect and the latter is what is usually called circulatory lift, which is given by L=−

π

ρ U 2 α (2b + U ∆t ) . 2 Taking the limit as ∆t goes to zero, one obtains the initial circulatory lift as LC =

LC (t = 0) = π ρ U 2 bα, which is half of the steady lift 2πρ U 2 bα . Thus, the initial value of Wagner’s circulatory lift can be determined from purely added mass considerations. Because LC is composed of lift due to circulation and some acceleration effects and since the latter is negligible at lower frequencies, Theodorsen function behaves like a low pass filter in the low frequency range (following the circulation transfer function) with a monotonically decreasing magnitude and phase, as shown in Fig. 4. Also, for low frequencies, it is fair to assume that the circulatory lift is due to circulation and its rate of change, as given by the approximation (11) and that of Lomax et al. (1952), shown in Fig. 6. On the other hand, since the acceleration terms are dominant in the high frequency range, the Theodorsen function departs from the circulation transfer function and follows an algebraic-type relation with the input (with half of the magnitude and zero phase difference), similar to the added mass effects which are proportional to the instantaneous input acceleration. In fact, the notion accentuated in this work that the circulatory lift (with its common definition) includes some acceleration (non-circulatory) contributions is physically plausible. Recall Theodorsen’s formulation that the non-circulatory flow takes care of the no-penetration boundary condition, but does not satisfy the Kutta condition. Then, Theodorsen added the circulatory flow field such that (i) the no-penetration boundary condition remains intact, (ii) Kelvin’s conservation of circulation is satisfied, and (iii) the superposition of the two flow fields (circulatory and non-circulatory) satisfy the Kutta condition at the trailing edge. This last requirement yields the governing integral equation for wake circulation. Therefore, this equation connects the two flow fields. In particular, the resulting lift force from the second flow field (the circulatory lift) depends on the non-circulatory flow (added-mass effects) through this equation representing the Kutta condition.

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Fig. 7. A comparison between the time history of the viscous added-mass lift (at R = 40,000) and the potential flow one, both normalized by the added-mass ma = πρ b2 and the maximum acceleration of the mid-chord point.

6. Circulation in a non-circulatory motion As we showed that the circulatory–non-circulatory classification is arbitrary and that the circulatory lift actually possesses some added-mass contribution, it is interesting to investigate whether a non-circulatory maneuver (a maneuver that is expected to generate no circulatory lift) induces loads due to circulation. To pursue such an investigation, we perform laminar numerical simulations of the Navier–Stokes equations on a dynamic mesh around a NACA 0012 airfoil with sharp trailing edge undergoing pitching motion hinged at the quarter-chord point in a quiescent fluid (no free stream). Details about the computational setup and the dynamic mesh can be found in our previous efforts (Rezaei and Taha, 2017; Taha and Rezaei, 2018, 2019). Since there is no free-stream, the Reynolds number is defined based on the ( )2 2 oscillation frequency ω as R = ωνb , which is related to the thickness δ of the Stokes layer as R = 2 bδ . The simulation is carried out using a 3◦ pitching amplitude and R = 40,000. According to the classical inviscid theory of unsteady aerodynamics (Wagner, 0000; Theodorsen, 1935; Von Karman and Sears, 1938), the expected lift is purely due to added mass effects, which is given by L(t) = LNC (t) =

1 4

πρ b3 α¨ (t)

for pitching around the quarter-chord point. Therefore, the generated lift force is proportional to the instantaneous acceleration with no phase lag. Fig. 7 shows a comparison between the computed lift time–history against the potentialflow one. Clearly, the computed viscous lift possesses a phase lag with respect to the acceleration α¨ in contrast to the potential-flow lift. It is quite interesting to conclude that even the added mass-effects possess non-trivial dynamics (lag), induced by viscosity. In fact, the snapshots of the vorticity contours at some instants during the cycle, shown in Fig. 8, point to the shedding of strong vortices from the airfoil trailing edge, which implies the development of circulation over the airfoil by virtue of conservation of circulation. Fig. 9 shows the time history of the bound circulation over the airfoil computed from our laminar viscous simulation at R = 40,000, along with the angular acceleration (representing the non-circulatory lift). It is interesting to point out that the phase difference between Γ and α¨ is 45◦ . From this discussion, it is concluded that viscosity induces bound circulation over the airfoil even in the absence of a free-stream. In other words, similar to the fact that the circulatory lift includes added-mass effects (even in potential flow), the non-circulatory lift seems to also enjoy some viscous circulatory contributions. Running this computational experiment at different frequencies (corresponding to different Reynolds number in the current definition) to construct the frequency response of the added mass contribution, we find that the potential flow theory predicts the magnitude (quadratic in frequency) quite well, as shown in Fig. 10. However, interestingly, we observe a phase difference from the potential flow results that is almost independent of frequency, quite similar to the flow behavior in the Stokes second problem (Batchelor, 2000, pp. 191–193; Lamb, 1932, pp. 619–623; Langlois and Deville, 2014, pp. 109–111). In fact, the only difference between the current problem and that of Stokes is the finite length of the considered airfoil in contrast to Stokes infinite plate. For more details about the viscosity-induced lag, the reader is referred to our recent effort (Taha and Rezaei, 2018, 2019) or the work of Othman et al. (2018).

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H.E. Taha and A.S. Rezaei / Journal of Fluids and Structures 93 (2020) 102868

Fig. 8. Vorticity contours at four time instants during a cycle of pure pitching in the absence of free stream at R = 40,000.

Fig. 9. Time history of the bound circulation over the airfoil computed from CFD at R = 40,000, along with the angular acceleration (representing the non-circulatory lift). The bound circulation is normalized as Γ˜ = ωΓb2 .

7. Effect of compressibility and viscosity We would like to conclude this paper by discussing the effect of compressibility and/or viscosity on the dynamics of unsteady lift. It should be noted that the Laplacian, which governs the potential flow dynamics, has no time derivative. That is, at each time step, the Laplacian in the velocity potential F is solved subject to the instantaneous boundary conditions dictated by the instantaneous airfoil motion. In other words, the velocity field at each time step corresponds to the instantaneous angle of attack, as if it were a steady problem (no memory effects). By definition, this approach is a quasi-steady one, although the dF /dt term in the unsteady Bernoulli’s equation captures the pressure variations necessary to accelerate the fluid from the velocity field at some time instant to that at the next time step (i.e., added mass contributions). In other words, no regard is given to the dynamics of the flow field to match the new boundary condition. It is assumed that such development happens instantaneously in potential flow. That is, the brute force solution of the Laplacian in the velocity potential subject to the no-penetration boundary condition and followed by application of the unsteady Bernoulli’s equation is not adequate to describe the flow dynamics; in fact, it completely ignores the flow dynamics. To better describe the flow physics, within the framework of potential flow, one must deliberately account for the wake effects by shedding vorticity from the sharp edges; the Kutta condition would then be used to determine the strength of this shed vorticity (see a comprehensive discussion on the applicability of Kutta condition to unsteady flows

H.E. Taha and A.S. Rezaei / Journal of Fluids and Structures 93 (2020) 102868

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Fig. 10. Computational results of the non-circulatory lift frequency response versus Theodorsen’s potential flow results. Since the computation is performed at zero free-stream, the frequency is presented in terms of the Reynolds number R = the Stokes layer thickness).

ωb2 ν

(twice the ratio between the semi-chord and

in our recent effort Taha and Rezaei, 2019). It is mainly the wake effect that captures the dynamics (i.e., causes lag in the development) of lift and circulation in potential flow. Accounting for viscosity and/or compressibility, however small they may be, will preclude such a singular behavior. The dynamics will no longer be governed by the Laplacian; the time-derivative will explicitly appear in the governing equation and viscosity and/or compressibility will affect the time-scale (e.g., the lag) of the lift and circulation development. Graham (1983) argued that the viscous effects will be significant over a very short time right after the impulsive start: in the order of U −1 bR1−2κ , where R is the Reynolds number. So, for practical trailing edge angles κ ≃ 2 and high Reynolds numbers, this time is very short, which suggests that the lift short-time (high-frequency) behavior may enjoy significant viscous effects. This fact is also supported in our recent effort (Taha and Rezaei, 2019) where we developed a viscous extension of Theodorsen’s potential-flow lift frequency response. It is shown that viscosity induces more phase lag to the lift dynamics, particularly at low Reynolds numbers and high frequencies. Graham (1983) also argued that compressibility would induce significant lag on the lift and circulation dynamics, unless the Mach number is not too small (smaller than 0.1). 8. Conclusion In this paper, we emphasize the dynamical-systems aspect of unsteady aerodynamics. That is, we consider the unsteady aerodynamics problem of a two-dimensional airfoil as a dynamical system whose input is the angle of attack (or airfoil motion) and output is the lift force. Based on this view, we discuss the lift evolution from a purely dynamical perspective through the step response (Wagner function), frequency response (Theodorsen function), transfer function, etc. In particular, we show that the circulatory lift dynamics has a non-zero high-frequency gain. Therefore its transfer function is not strictly proper, which points to the existence of a component in the circulatory lift that responds instantaneously without a lag: half of the steady lift is immediately generated (instantaneously developed). In contrast, the circulation transfer function is strictly proper with a zero high-frequency gain (i.e., acts as a low pass filter). We show the dynamical relation between the circulatory lift and circulation. It is found that the circulation follows the circulatory lift with a delay in the low frequency range and a fractional integrator in the high frequency range. Therefore, in the low-frequency range, it is fair to assume that the circulatory lift is lift due to circulation and its rate of change. However, in the highfrequency range, this is not true. By comparing the steady and unsteady versions of thin airfoil theory, we show that the circulatory–non-circulatory classification is arbitrary. We show that the circulatory lift includes some acceleration terms, which explain its high-frequency (instantaneous) response. On the other hand, we perform laminar numerical simulation of the Navier–Stokes equations on a dynamic mesh around NACA 0012 undergoing pitching motion in a quiescent fluid (no free stream) to show that such a non-circulatory maneuver (a maneuver that is expected to result in no circulatory lift) induces viscous circulation over the airfoil. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. CRediT authorship contribution statement Haithem E. Taha: Conceptualization, Formal analysis, Funding acquisition, Investigation, Methodology, Writing original draft. Amir S. Rezaei: Data curation, Software, Validation, Writing review & editing.

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Acknowledgments The authors would like to acknowledge the National Science Foundation grant CMMI-1846308. The first author is very thankful to the fruitful discussions with Prof. Saad Ragab at Virginia Tech on the physics of circulation development. References Batchelor, G.K., 2000. An Introduction to Fluid Dynamics. Cambridge university press. Beddoes, T.S., 1983. Representation of airfoil behavior. Vertica 7 (2), 183–197. Beddoes, T.S., 1984. Practical computation of unsteady lift. Vertica 8 (1), 55–71. Birnbaum, W., 1924. Der schlagflugelpropeller und die kleinen schwingungen elastisch befestigter tragfluegel. Z. Flugtech Motorluftschiffahrt 15, 128–134. Bisplinghoff, R.L., Ashley, H., Halfman, R.L., 1996. Aeroelasticity. Dover Publications, New York. Brown, C.E., 1954. Effect of leading-edge separation on the lift of a delta wing. J. Aeronaut. Sci. 21, 690–695. Brunton, S.L., Dawson, S.T.M., Rowley, C.W., 2014. 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