A New Expression for Quasi-Steady Effective Angle of Attack Including Airfoil Kinematics and Flow-Field Nonuniformity

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Citation: Xu, X. and Lagor, F. D., A New Expression for Quasi-Steady Effective Angle of Attack Including Airfoil Kinematics and Flow-Field Nonuniformity. AIAA SciTech Forum, 1782, 2020.

A New Expression for Quasi-Steady Eﬀective Angle of Attack Including Airfoil Kinematics and Flow-Field Nonuniformity

Xianzhang Xu∗ and Francis D. Lagor† University at Buﬀalo, The State University of New York, Buﬀalo NY 14260

The effective angle of attack of an airfoil is a composite mathematical expression from quasi-steady thin-airfoil theory that combines the geometric contribution to angle of attack with pitching and plunging effects. For a maneuvering airfoil, the instantaneous effective angle of attack is virtual angle that corresponds provide equivalent lift based on a steady, lift versus angle of attack curve. The existing expression for effective angle of attack depends on attached- flow, thin-airfoil, small-angle, and small-camber-slope assumptions. This paper derives a new expression for effective angle of attack that relaxes the small-angle and small-camber- slope assumptions. The new expression includes effects from pitching, plunging, and surging motions, as well as spatial nonuniformity of the flow. The proposed expression simplifies to the existing quasi-steady expression by invoking the appropriate assumptions. Further, the proposed expression leads to a replacement for the classic, zero-lift angle of attack equation for steady flow past a thin airfoil, which is compared to experimental values for cambered NACA airfoils. The new expression is also used in a problem for matching the effective angle of attack to design a pitching maneuver that emulates the effective angle of attack of a non-pitching airfoil encountering a transverse gust.

I. Nomenclature Greek Letters α, αÛ = angle of attack, angle of attack rate αeﬀ = eﬀective angle of attack β = side-slip angle γ = vortex sheet strength per unit length Γ = total circulation θ = angular coordinate along mean camber line θ˜ = dummy angle for angular identities θ0 = angular coordinate of evaluation location on mean camber line Û θg, θg = geometric angle of attack, pitch rate ξ = location along camber line ρ = density of air

Roman Letters a = coordinate of pitch axis in semi-chords An, Bn = Fourier series coeﬃcients b = semi-chord length, = c/2 c = chord length C = name of center of rotation point Cl = sectional lift coeﬃcient dz/dx = slope of the mean camber line G = name of center of mass point hÛ = plunge rate L = Gust width L0 = lift per unit length N = name of leading edge point (nose) of airfoil

∗Graduate Student, Department of Mechanical and Aerospace Engineering, AIAA Student Member †Assistant Professor, Department of Mechanical and Aerospace Engineering, AIAA Senior Member

1 O = name of origin point for inertial frame P = name of point along the camber line sÛ = surge rate V = constant, characteristic flow speed Vn = nondimensionalized flow component normal to the camber line V∞ = magnitude of freestream velocity V∞,n = magnitude of normal component of freestream velocity Vg = maximum velocity of gust encounter win = induced flow component win,n = component of induced velocity normal to the camber line

Vectors, Reference Frames, Components bˆx, bˆy, bˆz = basis vectors of the body frame eˆx, eˆz = basis vectors of frame I nˆ x, nˆ z = basis vectors of frame N N = Body-fixed reference frame at point N, the nose of the airfoil rP/O = position vector of point P relative to point O R = Rotating reference frame located at point C, the axis of rotation (s, h) = coordinates of point C in reference frame I I (u, v, w) = components of vG/O expressed in the body frame 0 0 0 I (u , v , w ) = components of vG/flow expressed in frame B, I = components of vP/flow expressed in frame R, I = components of vflow/P expressed in frame N (ux, uz) = external flow components in the inertial reference I I vG/flow, vP/flow = flow-relative velocity vectors I vG/O = inertial velocity of point G relative to point O wˆ x, wˆ y, wˆ z = basis vectors of the wind frame IωR = angular velocity of frame R relative to frame I W = Wind frame located at point G (x, z) = coordinates of point P in reference frame N (xC, zC ) = coordinates of point C in reference frame N

II. Introduction hin-airfoil theory has proven to be effective for the prediction of the aerodynamic forces exerted on airfoils with Tattached flow and small angles of attack. Steady thin-airfoil theory assumes that a sufficiently thin airfoil with a steady and attached flow can be represented by a thin sheet of vortices along its mean camber line, which acts as a streamline of the flow [1]. Quasi-steady thin airfoil theory, an adaptation of the steady theory, incorporates slowly varying pitch, surge, and plunge motions of the airfoil [2]. As an extension of quasi-steady thin-airfoil theory, the works of Theodorsen [3], Wagner [4], Sears [5], von Kármán [2], and others contributed to the unsteady thin-airfoil theory that examines airfoil motions with prominent acceleration terms and flow or wake unsteadiness. Theodorsen developed an unsteady lift model that includes the effects of added mass and wake vorticity [3]. Wagner proposed an unsteady aerodynamic model similar to Theodorsen’s that simulates the response of a step change in angle of attack [4]. Although many of these models have limiting assumptions for inviscid, incompressible, and attached flow, they are still widely used as fundamental tools in unsteady aerodynamics research [6]. Across the steady, quasi-steady, and unsteady thin-airfoil theories, a useful concept for comparing the lift generated by various airfoil motions is effective angle of attack. The effective angle of attack of an airfoil represents a composite view of the flow along the airfoil that considers various sources of flow-relative motion. Leishman [7] presented an effective angle of attack for a quasi-steady thin airfoil undergoing pitching and plunging maneuvers using the concept of virtual camber. Sedky et al. [8] included transverse gust effects into this expression with the use of virtual camber. Brunton et al. [9] investigated the lift generation of a pure plunging flat plate through 2D simulation and the effective angle of attack was applied based on Theodorsen’s model. Ramesh et al. [10] used a Leading Edge Suction Parameter (LESP), which is related the effective angle of attack through the zeroth Fourier coefficient of the airfoil’s bound

2 circulation, to modulate the shedding of vorticity at the leading edge in a discrete-vortex model. Several authors have also used the effective angle of attack or the LESP as an internal state variable for the wing in a dynamic model of the lift. Sedky et al. [8] incorporated the effective angle of attack a state variable in a dynamic Goman-Khrabrov (GK) model for a pitching wing. Narsipur et al. [11] similarly incorporated the related LESP quantity as a state variable in a GK model. Visbal and Garmann [12] used the effective angle of attack as a tool for interpreting the state of the wing during analysis of motions for pitch and plunge equivalency. Many of these works have used the effective angle of attack due to its ability to incorporate kinematic effects and its interpretability, even though applications with separated flow, large amplitude motions, or large amplitude gust encounters often violate the assumptions underlying the effective angle of attack derivation from thin-airfoil theory. Given the common use of effective angle of attack, this paper seeks to reduce the number of assumptions involved in its derivation. To this end, we remove the small-angle and small-camber-slope assumptions. We also perform a formal kinematic analysis to include all quasi-steady kinematic contributions (e.g., pitching, plunging, and surging motions) of the airfoil, as well as freestream nonuniformity (e.g., gusts). The result is a useful expression for effective angle of attack that can be easily applied to various airfoil motions without the additional step of a virtual camber calculation. In this work, a new expression is derived based on three modifications to the prior derivation: (i) we include a formal kinematic analysis for the motion of an arbitrary point on the mean camber line of the airfoil; (ii) we utilize an expression of angle of attack based on the relative velocity of the fluid and apply it local to the each point on the camber line of the airfoil; and (iii) we incorporate useful trigonometric identities to retain nonlinearities that would otherwise be removed by simplifying assumptions. These alterations produce an effective angle of attack expression that includes the effects of surging, plunging, and pitching motion with additional nonuniform freestream effects. The new expression has a reduced dependence on the small-angle and small-camber-slope assumptions. Although we do not explicitly invoke a small-angle assumption, the assumption of flow attachment implicitly limits the work to relatively small angles. Similarly, we remove the small-camber-slope assumption from the derivation, however, it still appears implicitly in the projection of the Cauchy integral of the bound circulation onto the horizontal axis. Nonetheless, our new expression is a useful extension of the previous expression for effective angle of attack from quasi-steady thin airfoil theory. In the special case of steady flow past a cambered airfoil, our expression provides a better prediction of the zero-lift angle of attack than the existing theory. Additionally, in contrast to other methods of calculating an effective angle of attack (e.g., see [7]), our approach does not require calculation of a virtual camber, thereby simplifying the calculation and allowing effective angle of attack to be applied to (physically) cambered airfoils. The outline of this paper is as follows: Section III reviews the effective angle of attack in classical thin-airfoil theory. Section IV provides a kinematic analysis of motion of an arbitrary point on the mean camber line of a thin airfoil. Section V uses the kinematics of the airfoil, an expression for angle of attack from aircraft flight dynamics applied locally, and trigonometric identities to derive a new expression for effective angle of attack. Section VI examines special cases of the expression, and Section VII uses the expression to create framework for applications that involve creating an airfoil motion profile that matches a desired effective angle of attack trajectory. Section VIII concludes the paper and discusses ongoing work.

III. Effective Angle of Attack in Thin-Airfoil Theory This section reviews thin airfoil theory as detailed by Anderson in [1] and the derivation of a prior expression for effective angle of attack as presented by Leishman in [7]. Thin-airfoil theory provides a potential-flow-based method of predicting the aerodynamic loading on an airfoil based on the freestream flow conditions and the geometric and kinematic characteristics of the airfoil. Thin-airfoil theory represents a sufficiently thin airfoil using a vortex sheet placed along the mean camber line, as shown in Fig. 1(a). For a thin airfoil with chord length c and with small camber, the camber line is close to the chord line. A helpful analytical approximation shifts the vortex sheet along the chord line so that the sheet strength may be expressed as γ = γ(x). The vortex sheet must satisfy the Kutta condition that the flow leaves smoothly at the trailing edge of the airfoil [1], which results in an end condition of γ(c) = 0. The vortex sheet acts as a streamline of the flow so that the velocity component normal to the camber line must be zero at all points along the camber line [1]. Determining the proper circulation-strength distribution to meet these requirements results in a total circulation that can be used to determine the lift force and pitching moment experienced by the airfoil [1]. Let win,n be the component of the flow velocity induced by the vortex sheet that is normal to the camber line (i.e., the induced flow component arising from vortex sheet elements along the camber line). For camber line to be a streamline, the normal component of the impinging flow and the normal component of the velocity induced by the vortex sheet

3 (a) (b) (c)

Fig. 1 a) Vortex sheet on the mean camber line. b) Geometry for ﬁnding the normal component of freestream velocity V∞,n. c) Velocity induced by the vortex sheet on the chord line. (Figures adapted from [1].)

must cancel such that V∞,n + win,n = 0, (1) where V∞,n is the normal component of the freestream velocity that can be found by inspection of Fig. 1(b). The slope of camber line segment is dz/dx, and Fig. 1(b) shows that −1 dz V∞ = V∞ sin α + tan − . (2) ,n dx

Classical thin-airfoil theory makes a small-angle assumption, sin θ˜ ≈ tan θ˜ ≈ θ˜, to simplify (2), such that dz V∞ = V∞ α − , (3) ,n dx and simpliﬁes calculations by placing the vortex sheet on the chord line (i.e., assuming the airfoil’s camber is small). Figure 1(c) illustrates the calculation of the velocity component win that is induced by the vortex sheet and is normal to the chord line. The velocity dwin at point x induced by the sheet element at point ξ is given by γ(ξ)dξ dw = − . (4) in 2π(x − ξ)

The velocity win(x) at point x induced by the vortex sheet can be obtained by integration from the leading edge at x = 0 to the trailing edge at x = c such that ¹ c γ(ξ)dξ win(x) = − . (5) 0 2π(x − ξ) Under the assumption that win ≈ win,n and using (1) to equate the normal velocity components to enforce the camber line as a streamline leads to the Fundamental Equation of Thin Airfoil Theory [1] dz 1 ¹ c γ(ξ)dξ V∞ α − = . (6) dx 2π 0 x − ξ Equation (6) states that normal flow component at point x equals to the induced velocity from the vortex sheet due to the integration of all vortex elements along the chord. If the flow condition V∞, geometric flow angle α, and the camber slope dz/dx are specified in (6), the vortex sheet strength γ(ξ) can be found to enforce the Kutta condition and satisfy the streamline condition present in (6). However, the left-hand side of the equation is derived based on small-angle and small-camber-slope assumptions. Section V re-visits this derivation and removes these assumptions from this step. Parameterizing by an angular coordinate θ under the change of variables c x = (1 − cos θ), (7) 2 aids in evaluating the integrals involved in solving (6) for γ(ξ) and converts (6) to dz 1 ¹ π γ(θ) sin θdθ V∞ α − = . (8) dx 2π 0 cos θ − cos θ0

4 The integral equation (8) can be solved for the circulation distribution [1]

∞ ! 1 + cos θ Õ γ(θ) = 2V∞ A + An sin nθ . (9) 0 sin θ n=1 Plugging in (9) into (8) and evaluating using the trigonometric integral identities [1]

¹ π sin nθ sin θdθ = −π cos nθ0, (10) 0 cos θ − cos θ0

and,

¹ π cos nθdθ π sin nθ = 0 , (11) 0 cos θ − cos θ0 sin θ0 leads to the camber slope represented as a cosine series expansion[1]

∞ dz Õ = (α − A ) + A cos nθ . (12) dx 0 n 0 n=1 Í∞ In Fourier analysis, functions of the form f (θ) = B0 + n=1 Bn cos nθ, have Fourier coeﬃcients that can be found by taking the inner product of both sides with another cosine function and utilizing the orthogonality of cosine harmonics, leading to [1]

1 ¹ π B0 = f (θ)dθ (13) π 0 2 ¹ π Bn = f (θ) cos nθdθ. (14) π 0 Similarly, the Fourier coeﬃcients of (12) are

1 ¹ π dz A0 = α − dθ0 (15) π 0 dx 2 ¹ π dz An = cos nθ0dθ0. (16) π 0 dx The remainder of the derivation of the lift coeﬃcient from thin-airfoil theory follows from integration of the circulation distribution and application of the Kutta-Joukowski theorem. The total circulation of the vortex sheet on the camber line is [1] ¹ c c ¹ π Γ = γ(ξ)dξ = γ(θ) sin θdθ, 0 2 0 and plugging in the assumed form of γ yields A1 Γ = πcV∞ A + . (17) 0 2

By the integral identity (10), note that only the A0 and A1 Fourier coeﬃcients survive. Using the Kutta-Joukowski theorem, the coeﬃcient of lift per unit span can be expressed [1] 0 L ρV∞Γ A1 Cl = = = 2π A + . (18) 1 2 1 2 0 2 2 ρV∞c 2 ρV∞c

The linear relationship Cl = 2πα can be compared to (18) and the terms within the parentheses of (18) can be defined as an effective angle of attack A α = A + 1 . (19) eff 0 2

5 Substitution of (15) and (16) leads to the final expression for the lift coefficient in steady, thin-airfoil theory, [1] 1 ¹ π dz Cl = 2π α + (cos θ0 − 1) dθ0 , (20) π 0 dx | {z } αeff in which the integral term incorporates the effect of camber.

For a quasi-steady theory, Leishman [7] gives a textbook derivation of the expression for effective angle of attack for a symmetric, thin airfoil of semi-chord length b that is undergoing pitching and plunging motions. The kinematic motions of the symmetric airfoil introduce relative flow velocities along the chord that result in lift that is identical to the lift produced by a virtual airfoil that may have camber (known as virtual camber) and a geometric angle of attack. The kinematic effects due to plunge-rate hÛ and pitch-rate αÛ prescribe a distribution of the relative velocity along the chord that can be used to find the virtual camber distribution. Equations (15), (16), and (18) from thin-airfoil theory are then applied to the virtually cambered airfoil to yield a quasi-steady effective angle of attack [7]

hÛ 1 αÛ αeff = α + + b − a , (21) V∞ 2 V∞ where the pitch axis is located a semi-chords from the mid-chord. Sedky et al. [8] use the idea of virtual camber to include effects from a transverse gust Vg, providing an expression that adds onto (21), such that Û π h 1 αÛ 1 ¹ Vg(θ0) αeff = α + + b − a + (cos θ0 − 1) dθ0. V∞ 2 V∞ π 0 V∞

IV. Local Angle of Attack for a Point on the Camber Line The motion of an airfoil alters the inertial velocities of points on the camber line, which also changes the local velocity relative to the flow and the associated local angle of attack. This section derives the kinematics for an arbitrary point P on the camber line during the two-dimensional motion of an airfoil. Using these kinematics, this section then calculates the local angle of attack at point P. The notational conventions in this section are based on [13] for engineering dynamics and [14] for aircraft flight dynamics. Let B = G, bˆx, bˆy, bˆz be a body-fixed reference frame (i.e., the body frame) with unit vectors bˆx, bˆy, and bˆz, that is located at the center of mass G of an aircraft flying as shown in Fig. 2(a). In aircraft flight dynamics (e.g., see [14]), it is useful to also construct a reference frame located at the center of mass G that is known as the wind frame W = G, wˆ x, wˆ y, wˆ z . The wind frame has the wˆ x direction aligned in the opposing direction of the relative wind experienced by the aircraft, causing the aerodynamic forces of lift, drag, and side force act along the reference directions in this frame. To construct the wind frame, rotate the body frame through an angle of attack α about the −bˆy axis of the body frame, followed by a rotation through a side- slip angle β about the −bˆz direction. These rotations are left-handed rotations (i.e., rotations about the negative direction of the axis in the right-handed sense) to ensure matching of the definitions of these angles between flight dynamics and historical aerodynamics literature [14]. Let I = (O, eˆx, eˆy, eˆz) be an inertial reference frame fixed in space at point O with unit vectors eˆx, eˆy, and eˆz. The aerodynamic characteristics of the aircraft’s dynamics are determined by the velocity of the aircraft relative to the surrounding air, which is given by [14] I I I vG/flow = vG/O − vflow/O, (22) I where vG/O is the inertial velocity of the aircraft’s center of mass and vflow is the velocity of the wind. The left superscript I indicates that the vector was derived using time differentiation with respect to the inertial frame. Using the body frame B, the relative velocity components can be expressed as [14]

u0  u u       flow  [I ]  0    −   vG/flow B = v  = v  vflow  , (23)  0     w  w wflow   B   B   B where u, v, and w are the aircraft’s velocity components and uflow, vflow, and wflow are the freestream wind components. The notation [v]B indicates that the array entries are components of the vector v expressed in frame B.

6 (a) (b)

Fig. 2 a) Aircraft ﬂight dynamics conventions for the body frame and the wind frame. b) Thin airfoil geometry and relevant reference frames.

Based on the geometry of Fig. 2(a), the angle of attack is given by the components of (23) as [14] 0 −1 w −1 w − wflow α = tan 0 = tan . (24) u u − uflow The angle of attack (24) includes both the motion of the vehicle and the freestream air flow. In aircraft flight dynamics, this expression applies at the center of mass G of the aircraft and is useful in deriving the aircraft’s equations of motion using the Euler’s first law for rigid body motion, in which the aircraft is abstracted as a point mass located at the center of mass. In the following, the point-wise applicability of (24) enables application at individual points along the camber line of an airfoil to produce a local angle of attack at each point on the camber line. If the air is quiescent or the wind components are known from a model or measurements, only determining the u and w components of the inertial velocity in (24) is needed. The uflow and wflow components may come from impinging flow due to wind or represent flow components from wake effects, downwash/upwash, or induced flow from nearby coherent flow structures. To derive the kinematics of a point along the mean camber line of an airfoil, consider a thin airfoil that can be represented by its mean camber line in Fig. 2(b). Figure 2(b) presents reference frames that are relevant to the construction of the kinematics. Let R = (C, bˆx, bˆy, bˆz) be a reference frame that translates and rotates with the airfoil and is located at point C, which is the center of rotation for pitching maneuvers. Frame R is aligned with the body frame B of the vehicle but the center of rotation C may be offset from the center of mass G. The orientations of frames R and I match the common convention in aircraft flight dynamics, in which the body-x direction extends out of the nose of the aircraft and the body-z direction extends downward from the fuselage. For convenience, we also define reference frame N = (N, nˆ x, nˆ y, nˆ z) located at point N, which is the nose of the airfoil. The orientation of frame N matches the orientation often used in the aerodynamics literature (e.g., see [1, 7]), although it is sometimes located at the midchord (e.g., see [15]). Let rP/O be the position of an arbitrary point P on the camber line of the airfoil. We seek the inertial velocity of point P so that we can replace point G with point P in (22) and (23) for calculation of a local angle of attack at point P using (24). Vector addition gives the position of point P relative to point O

rP/O = rC/O + rP/C, (25) where point C is the center of rotation for pitching maneuvers. Since many of the thin-airfoil theory calculations occur in frame N, consider the position of point P relative to the frame N. From the geometry in Fig. 2(b), we have that

rP/C = rP/N − rC/N . (26) Taking the inertial time derivative of (25) gives the inertial velocity of point P, I I I vP/O = vC/O + vP/C . (27) I The contribution of vP/C to (27) results from pitching of the airfoil. Since frame R rotates with respect to frame I with angular velocity IωR, the Transport Equation [13] I R I R vP/C = vP/C + ω × rP/C, (28)

7 R is needed to diﬀerentiate rP/C . Note that vP/C , which is the time rate of change of the vector rP/C in frame R, is zero since point P is stationary in R. Collecting (26), (28), and (27) provides the inertial velocity of the point P as I I I R vP/O = vC/O + ω × rP/N − rC/N . (29)

We select coordinates for the position vectors in this problem by noting that (x, z) from Fig. 1 are the coordinates of rP/N expressed in frame N and assigning (xC, zC ) as the coordinates of rC/N expressed in frame N. The vector rP/C can then be written in coordinates as rP/N − rC/N = (x − xC ) nˆ x + (z − zC ) nˆ z . (30)

Let (s, h) be the coordinates of rC/O expressed in frame I so that rC/O = seˆx + heˆz, and the inertial velocity of the I Û Û point C relative to O is vC/O = sÛeˆx + heˆz, where sÛ and h represent the surge and plunge rates of the airfoil, respectively. To calculate the angular velocity vector IωR in (28), we first examine the orientation of the airfoil. The chord line of the airfoil is oriented at an angle θg, which describes the rotation angle of frame R relative to frame I. The orientation angle θg is an Euler angle that describes the pitch attitude of the airfoil, although we sometimes refer to θg as a geometric angle of attack, since it corresponds with the angle of attack from traditional testing for a non-maneuvering airfoil in the presence of a freestream flow in the −ex direction. The local angle of attack is formally specified by (24) Û and involves the freestream wind components as well as the motion of the airfoil. The pitch rate is θg, and the angular I R Û Û velocity of frame R with respect to I becomes ω = θg eˆy = θg bˆy. Substitution into (29) gives I I Û vP/O = vC/O + θg bˆy × − (x − xC ) bˆx − (z − zC ) bˆz , Û Û Û = sÛeˆx + heˆz − θg (z − zC ) bˆx + θg (x − xC ) bˆz .

Transforming the inertial unit vectors using eˆx = cos θg bˆx + sin θg bˆz and eˆz = − sin θg bˆx + cos θg bˆz provides

I Û Û Û Û vP/O = sÛ cos θg − h sin θg − θg (z − zC ) bˆx + sÛ sin θg + h cos θg + θg (x − xC ) bˆz, (31) Û which is the inertial velocity of point P due to the two-dimensional kinematic motions of pitching at a rate of θg about point C, plunging at a rate of hÛ, and surging at a rate of sÛ. Replacing G with P in (22) (and allowing the slight abuse of notation in which (u0, w0) now represent the velocity components of point P instead of point G), the relative velocity components become 0 Û Û u = sÛ cos θg − h sin θg − θg (z − zC ) − uﬂow, (32) 0 Û Û w = sÛ sin θg + h cos θg + θg (x − xC ) − wﬂow. (33)

I If convenient, the external ﬂow vector vﬂow/O may be expressed in the inertial frame with components ux and uz by

uﬂow = ux cos θg − uz sin θg, (34)

wﬂow = ux sin θg + uz cos θg. (35)

Many aerodynamic calculations from thin-airfoil theory are performed in frame N instead of frame B. Moreover, I I the relative velocity vflow/P is often considered instead of the vP/flow, which is used in the construction of (24). The relationship " 0 # hI i hI i u vP/flow = vflow/P = , (36) B N w0 N shows that (u0, w0) are the pertinent relative velocity components for the calculation of the local angle of attack in either convention. Plugging the u0 and w0 components from (32) and (33) into (24) yields the local angle of attack Û Û ! sÛ sin θg + h cos θg + θg (x − xC ) − w α = −1 flow , tan Û Û (37) sÛ cos θg − h sin θg − θg (z − zC ) − uflow which applies to an arbitrary point P on the camber line of a maneuvering airfoil in a nonuniform freestream.

8 V. A New Expression for the Eﬀective Angle of Attack of an Airfoil This section derives a new expression for eﬀective angle of attack that does not require the calculation of virtual camber. The new expression includes pitch, plunge, and surge motions of the airfoil, as well as freestream nonuniformity. From thin-airfoil theory, the induced velocity component in the nˆ z direction is given by (5) and the change of variables (7) as 1 ¹ π γ(θ) sin θdθ win(θ0) = − . (38) 2π 0 (cos θ − cos θ0)

Using win ≈ win,n and win,n = −V∞,n leads changes the expression to be in terms of the normal velocity component

1 ¹ π γ(θ) sin θdθ V∞,n(θ0) = . (39) 2π 0 (cos θ − cos θ0) The derivation in Section III proceeds by assuming a functional form of the circulation distribution γ(θ) to satisfy (39). A key observation that allows for the removal of assumptions is that the form of the circulation distribution that solves the integral equation (39) is unrelated to the small-angle and small-camber-slope assumptions made on the left-hand-side of (8). Even though these assumptions and the form of γ(θ) are often presented concurrently in the thin-airfoil theory derivation (e.g., see [1]), circulation distribution (9) solves the integral equation (39) and yields a cosine series representation of V∞,n(θ0), and this process does not actually require simpliﬁcations of V∞,n(θ0). This observation is widely known and has previously been utilized by several authors (e.g., see [10, 15–18].) The circulation distribution (9) in thin-airfoil theory corresponds to a cosine series representation of the ﬂow impinging normal to the camber line

∞ ! Õ V∞,n(θ) = V∞ A0 − An cos nθ , (40) n=1 which is found by inserting the circulation distribution (9) into the right-hand side of (39). Under a different convention of the angular coordinate and Fourier coefficients, such that φ = θ − π and γn = V∞ An, the authors in [15] equivalently Í∞ represent the right-hand side of (40) as n=0 γn cos (nφ). In the problem considered in this paper, the freestream flow can be spatially nonuniform and the airfoil can move freely. Hence, we replace the symbol V∞,n by Vn to reflect the fact that the freestream is less well-defined. Vn(θ0) represents the local, relative flow component that is normal to the camber line at the location θ0. Let V is a positive, constant, characteristic speed associated with the problem (e.g. the steady speed of a moving airfoil, or a spatially averaged wind speed for a stationary airfoil in an nonuniform flow) and define Vn(θ) = Vn(θ)/V to be the nondimensional flow component impinging perpendicular to the camber line. Nondimensionalizing Vn(θ) by V ensures the subsequent expressions for the Fourier coefficients will be nondimensional. In the proposed theory, the relative flow speed (i.e., V∞ from the traditional theory) is allowed to vary, so a constant characteristic speed is needed for nondimensionalization. Under these notational changes, we can and re-perform the thin-airfoil theory derivation and express the Fourier coefficients and effective angle of attack in terms of Vn. Let (40) be rewritten as

∞ ! Õ Vn(θ) = V A0 − An cos nθ . (41) n=1

The condition of no ﬂow across the camber line and the assumed form of Vn in (41) leads to the integral equation

∞ ! Õ 1 ¹ π γ(θ) sin θdθ V A − An cos nθ = . (42) 0 2π (cos θ − cos θ ) n=1 0 0

Assume a circulation distribution γ(θ) of the same form as (9), exchanging only V in place of V∞, so that

∞ ! 1 + cos θ Õ γ(θ) = 2V A + An sin nθ . (43) 0 sin θ n=1

9 Plugging in (43) verifies that this choice satisfies the integral equation (42). Fourier analysis provides the Fourier coefficients through (13) and (14), which become 1 ¹ π A0 = Vn(θ0)dθ0, (44) π 0 2 ¹ π An = − Vn(θ0) cos nθ0dθ0. (45) π 0 Noting the effective angle of attack in (19) remains unchanged by the choices made in this section, we substitute and plug in (44) and (45) for the effective angle of attack 1 ¹ π αeff = − Vn(θ0)(cos θ0 − 1)dθ0. (46) π 0

To obtain an expression for Vn to insert in (46), recall that Vn(θ) = Vn(θ)/V, utilize the geometry shown in Fig. 1(b), I replace V∞ with k vﬂow/P k, and insert the expression for angle of attack (24) into (2) to yield

p 0 2 0 2 0 (u ) + (w ) −1 w −1 dz Vn = sin tan + tan − . (47) V u0 dx Invoking the trigonometric identity

sin(θ˜1 − θ˜2) = sin θ˜1 cos θ˜2 − cos θ˜1 sin θ˜2, (48) transforms (47) into w0 dz w0 dz w0 dz sin tan−1 + tan−1 − = sin tan−1 cos tan−1 − cos tan−1 sin tan−1 . u0 dx u0 dx u0 dx Note that the derivation in Section III retains the angle of attack α. However, the presence of the arctangent function that results from the use of the local angle of attack expression (24) enables use of the following trigonometric identities: θ˜ 1 sin tan−1 θ˜ = √ , and cos tan−1 θ˜ = √ , (49) 1 + θ˜2 1 + θ˜2 which leads to w0 dz p 0 − ( 0)2 ( 0)2 u dx Vn = u + w s . (50) 2 w0 2 dz V 1 + u0 1 + dx

Under the mild assumption that u0 > 0 (to avoid having to consider the sgn(u0)), (50) can be rewritten as 0 0 dz w − u dx Vn = . (51) r 2 dz V 1 + dx Inserting (51) into (46) leads to our expression for the eﬀective angle of attack,

¹ π 0 0 dz 1 w − u dx αeff = − (cos θ0 − 1) dθ0, (52) π r 2 0 dz V 1 + dx where 0 Û Û u (θ0) = (sÛ − ux) cos θg − h − uz sin θg − θg (z (θ0) − zC ), (53) 0 Û Û c w (θ ) = (sÛ − ux) sin θg + h − uz cos θg + θg (1 − cos θ ) − xC . (54) 0 2 0 Expressions (53) and (54) are based on (32) and (33) after substituting the inertial frame components of the wind and the change of variables (7). The freestream components ux and uz may also be functions of θ0 if there is spatial nonuniformity in the surrounding flow field, such as in the case of a gust encounter.

10 VI. Special Cases This section examines implications of the proposed effective angle of attack expression for several applications, including a maneuvering symmetric airfoil, a thin airfoil encountering a transverse flow, and a cambered airfoil in steady, uniform flow. In addition to showing that the proposed expression (52) properly encompasses the prior effective angle of attack expression (21), this section presents a new expression for the zero-lift angle of attack for a cambered thin airfoil that replaces the classical expression.

A. Effective Angle of Attack for Maneuvering Symmetric Airfoil We consider whether the proposed expression for effective angle of attack (52) properly encompasses the previous effective angle of attack expression (21) for a pitching and plunging thin airfoil under the appropriate simplifying assumptions. The example in [7] (see pg. 430) considers a thin airfoil without camber (i.e., dz/dx = 0) in a freestream of flow speed V = V∞. Plugging into (52) leads to

1 ¹ π w0 αeﬀ = − (cos θ0 − 1) dθ0. (55) π 0 V∞

The example in [7] considers the relative ﬂow between the motion of the airfoil and the freestream and V∞, so we may assume that incident wind is in the inertial eˆx direction with ux = −V∞ and the airfoil is not surging forward (i.e., sÛ = 0). Û The thin airfoil has a geometric angle of attack θg = α and pitches at a rate of θg = αÛ about a point on the chord line Û located at (xC, zC ) = (b + ab, 0). The airfoil has a plunge rate of h. Plugging these expressions into (52) through (54), and using the fact that c = 2b gives

¹ π Û 1 V∞ sin α + h cos α + αÛ (b (1 − cos θ0) − b − ab) αeﬀ = − (cos θ0 − 1) dθ0, π 0 V∞ Û ¹ π ¹ π 1 V∞ sin α + h cos α − αÛ (b + ab) αÛ b 2 = − (cos θ0 − 1) dθ0 + (cos θ0 − 1) dθ0. π V∞ 0 πV∞ 0 The two integrals shown above can be evaluated analytically to yield −π and 3π/2, respectively, from left to right. Continued calculation reduces the eﬀective angle of attack to

hÛ αÛ (b + ab) 3bαÛ αeﬀ = sin α + cos α − + V∞ V∞ 2V∞ hÛ 1 αÛ = sin α + cos α + b − a . (56) V∞ 2 V∞

Equation (56) is the effective angle of attack for a flat, thin airfoil pitching about ab and plunging at a rate of hÛ. The first term is a geometric angle of attack contribution. The second term is a plunge-rate contribution, and the third term is a pitch-rate contribution. Note that if the small angle assumption is applied so that sin α ≈ α and cos α ≈ 1, we successfully recover the traditional effective angle of attack expression (21).

B. Angle of Attack Contribution for Transverse Flow Consider the scenario of a flat plate or symmetric airfoil (i.e. dz/dx = 0) in a steady freestream, at a constant geometric angle of attack, and encountering a transverse flow (e.g., see [8]). The need to incorporate the effect of a transverse flow in an angle of attack calculation often occurs in rotorcraft research, in which rotor blades experience downwash [15] or in fixed-wing flight applications in the presence of three-dimensional effects, such as wing-tip vortices −1 [1]. A common practice is to add a local angle of attack contribution tan (w/V∞) due to the transverse flow component w to the geometric angle of attack, yielding [8] −1 w αeff = θg + tan . (57) V∞

Peters et al. [15] note that although (57) is commonly implemented, it is known to be incorrect, because at zero 2 2 −1 geometric angle of attack, the lift should be proportional to 2πρbV∞ (w/V∞), not 2πρbV∞ tan (w/V∞).

11 Û Û Let the freestream be ux = −V∞ and take V = V∞. For this problem, we have sÛ = h = θg = 0. If the transverse ﬂow component uz is evaluated at a single, representative location, then substitution into (52) leads to

uz αeﬀ = sin θg − cos θg. (58) V∞

Equation (58) is the correct version of (57), where w = −uz cos θg due to the change of reference frames. The −1 transverse-ﬂow contribution to αeﬀ is (w/V∞), not tan (w/V∞), which agrees with the remark of Peters et al. [15].

C. Zero-Lift Angle for a Thin Airfoil Consider the steady ﬂow past a thin airfoil with camber slope dz/dx at a ﬁxed geometric angle of attack θg in a uniform freestream in the horizontal direction. The zero-lift angle from thin-airfoil theory is [1] 1 ¹ π dz αL=0 = − (cos θ0 − 1) dθ0. (59) π 0 dx

We find the zero-lift angle for the present theory by setting Cl = 2παeff = 0, where αeff is specified by (52), We also set Û Û ux = −V∞ and uz = sÛ = h = θg = 0 and solve for the geometric angle of attack θg at zero lift

π ∫ −1 dz ( θ − ) dθ 0 sin tan dx cos 0 1 0 θ = −1 © ª . g,L=0 tan π ® (60) ∫ −1 dz ( θ − ) dθ ® 0 cos tan dx cos 0 1 0 « ¬ Note that (60) does not readily simplify since the camber slope dz/dx is a function of the variable of integration θ0. Equation (60) replaces the classical result given in (59). For a more accessible comparison to existing results from thin-airfoil theory, consider instead the value of the sectional lift coefficient at zero geometric angle of attack. Setting θg = 0 within Cl = 2παeff, where αeff is specified by (52) results in ¹ π 1 −1 dz Cl θg = 0 = 2π − sin tan (cos θ0 − 1) dθ0 . (61) π 0 dx Equation (61) is directly comparable to the result from thin-airfoil theory that can be obtained by inserting (59) into Cl = 2π(α − αL=0) for α = 0. From this comparison, the two equations are identical under the approximation that dz dz sin tan−1 ≈ . dx dx

Equation (61) shows that the proposed theory of this paper encompasses the steady thin-airfoil theory results, but does so without invoking a small-camber-slope assumption. Table 1 presents the results of a comparison of thin-airfoil theory in (59), the present theory in (60), and experimental data obtained from the literature in [19] for several four-digit NACA airfoils. The equations for the camber lines of the

Table 1 Comparison of the zero-lift angle for four-digit NACA airfoils.

NACA airfoil Thin-airfoil theory in (59) Proposed theory in (60) Experiment in [19] 0012 0.000 0.000 0.0 2412 -2.077 -2.076 -2.0 4412 -4.155 -4.142 -4.1 6412 -6.232 -6.191 -6.1 6712 -9.130 -8.787 -7.4 8318 -7.672 -7.636 -7.4

four-digit NACA airfoil series can be found in [20]. Table 1 does not included a comparison to the data of [15], because Peters et al. [15] include additional approximations related to the truncation of a Fourier series that must be ﬁt to the camber line of the airfoil— these approximations complicate the comparisons. However, we note that the proposed expression (60) has the advantage that it does not require ﬁtting a Fourier series to the camber line prior to calculation.

12 The results in Table 1 show that the proposed theory produces zero-lift angle estimates that lies closer to the experimental values in all cases. For the very small camber, such as NACA 2412 airfoil, there is very little difference. As the maximum camber grows and moves towards the tail of the airfoil, such as in the cases of NACA 6412 and 6712 airfoils, the difference between the estimates is more apparent. The case of the NACA 6712 airfoil shows that if the location of maximum camber moves very far aft on the chord, both theories have difficulty predicting the true value.

VII. Effective Angle of Attack Matching This section presents the use of the proposed effective of attack expression for creation of a motion profile for a maneuvering airfoil to mimic a specified lift response. In particular, this section examines the how to match the effective angles of attack for a non-maneuvering airfoil that encounters a transverse, sine-squared gust and another, freely maneuvering airfoil that does not encounter a gust. Creating motion profiles that can mimic aerodynamic responses to unsteady flow phenomena, such as gust encounters, is important because development of unsteady aerodynamic experimental setups can be challenging [21]. Additionally, gust encounters are highly relevant to the current study of unsteady aerodynamics research [22, 23]. Consider a NACA 2412 cambered airfoil traveling with constant, horizontal speed sÛ = 0.4m/s at a constant geometric ◦ angle of attack θg,0 = 0 and encountering a transverse, sine-squared gust at t = 0, with an upwash profile that can be evaluated according to [8] ( 2 stÛ −x stÛ −x Vmax sin L π if 0 ≤ L ≤ 1, uz(x, t) = (62) 0 otherwise, where Vmax is the maximum speed of the gust and L is the gust width. Note that the piecewise construction of this function permits the evaluation of gust velocity along the chord with portions of the airfoil outside of the gust region. Consider also an additional airfoil that does not encounter a gust but tries to mimic the effect of the gust by pitching about the quarter chord, such that xC = 0.25c. The pitching airfoil has the same surge rate sÛ and the same initial geometric angle of attack θg,0. By letting Vmax = 0.2sÛ and assuming a chord of c = 0.1 m, the simulated gust encounter corresponds to gust ratio GR = Vmax/sÛ = 0.2 at a Reynolds number in water of approximately Re = 20, 000. To perform matching of the effective angle of attack, we construct the following constrained optimization problem that the uses the effective angle of attack expression proposed in this paper: Find the optimal (open-loop) control input u that solves ¹ tf 2 1 gust pitch min J(u) = α (t) − α θg, u dt (63) u eff eff 2 t0 Û subject to θg = u with θg(0) = θg,0. Û The control input u commands the pitch rate θg. Also note that the effective angle of attack for the gust encounter gust( ) ( ) / αeff t is only a function of time since it can be calculated using the known gust profile. The z x and dz dx terms in αeff correspond to the appropriate NACA airfoil equation for the mean camber line of a NACA 2412 airfoil, which can p 2 2 be found in [20]. The characteristic speed V for this problem is set to V = sÛ + Vmax. For the pitching airfoil, the pitch Û effective angle of attack αeff is a function of the airfoil motion variables θg and θg = u. To solve the constrained optimal control problem (63) computationally, we first assume a Fourier-series parameterization of the control input [24] N Õ u(t) = C2n−1 sin(nωt) + C2n cos(nωt), (64) n=1 where ω = 2π/T. The parameterization in (64) allows us to restrict the space of possible solutions for u(t) from an infinite-dimensional space of continuous curves on the interval [0, T] to the finite dimensional space of Fourier coefficients. That is, given N Fourier coefficients, a continuous curve u(t) is completely specified on the interval [0, T]. Larger values of N allow for more complex curves u(t). In the example we provide in this section, we select N = 6, which was found to provide sufficiently complex control input signals. For a pre-specified control u(t), the differential equation constraint in (64) can be readily integrated and plugged into the cost function to eliminate the constraint, yielding an unconstrained problem. We solve the unconstrained optimization problem using the standard numerical optimization function fminunc in MATLAB© [25]. The optimization solver appeared to occasionally get stuck in local minima, indicating that the optimization problem (64) may be nonconvex. To address this problem, we considered various random values of initial guesses

13 (a) (b)

Fig. 3 a) Eﬀective angles of attack for an airfoil encountering a gust and an airfoil mimicking a gust encounter through pitching. The dashed red line shows the optimal eﬀective angle of attack for the pitching airfoil. (Iterations of the optimization are suppressed for clarity.) b) Pitch rate input calculation during the matching optimization. Thin lines show the iterations converging towards the optimal solution (shown as a red dashed line).

for the Cn Fourier coefficients and separately solved the optimization problem from each initial guess. For the results of these solves, the solution with the lowest cost J was selected. For the initial guess of C1 = 0.0559 and Cn = 0 for n = 2,..., 12, the results of the optimization are shown in Fig. 3. This initial guess for the Cn Fourier coefficients resulted in an effective angle of attack for the pitching airfoil that was close enough to the effective angle of attack for the airfoil encountering a gust that the optimization routine was able to accurately and indistinguishably match the αeff curves, as shown in Fig. 3(a). The optimal solution is given by the Fourier coefficients (0.0956, −0.0217, −0.0004, 0.0004, 0.0008, −0.0001, 0.0008, 0.0000, 0.0011, 0.0000, 0.0013, 0.0000). The resulting control signal u(t) is shown in Fig. 3(b) as a dashed red line and consists of a pitch-up then pitch-down maneuver. Although the effective angle of attack matching problem of this section is focused on emulation of a gust encounter, the provided framework that utilizes the proposed effective angle of attack expression (52) and an optimal control formulation similar to (63) is general and applicable a various other problems of lift-force equivalence. For example, there has been significant work on pitch-plunge equivalence (e.g., see [12]) as well as more general, three degree-of-freedom motions for gust encounter emulation (e.g., see [21]). In ongoing work, we are using the proposed framework for analysis of these problems and others on lift-force equivalence.

VIII. Conclusion The paper presents a novel expression for a quasi-steady effective angle of attack from thin-airfoil theory that does not utilize the small-angle approximation or small-camber-slope approximation and includes kinematic effects related to pitching, plunging, and surging of the airfoil, as well as freestream nonuniformity. The method of obtaining the expression involved use of the local angle of attack due to the relative flow and trigonometric identities so that these two approximations were not needed. Assumptions of attached flow and a thin airfoil still remain. The resulting expression properly encompasses existing expressions for effective angle of attack and also provides a new expression for the zero-lift angle for cambered thin airfoil in steady flow conditions. Predicted values of the zero-lift angle for cambered airfoils are compared to experimental data obtained from literature, showing good agreement and a modest improvement over the existing expression from steady, thin-airfoil theory. This paper also applies the proposed effective angle of attack expression to a problem of emulating a transverse gust encounter using a pitching airfoil. The results of the proposed optimization framework provide an accurate solution for matching the effective angle of attack for an airfoil encountering a gust and another that pitches to mimic such an encounter, subject to the assumptions implicit in the effective angle of attack derivation. In ongoing work, we are extending the kinematic analysis of this paper to

14 examine unsteady motion eﬀects, and we are applying the eﬀective angle of attack matching framework to the study of pitch-plunge equivalence and the use of additional degrees of freedom in the gust emulation problem.

Acknowledgments The authors gratefully acknowledge valuable discussions with Mr. Girguis Sedky, Dr. Matthew Ringuette, Dr. Anya Jones, Dr. Jeﬀ Eldredge, Dr. David Williams, and Dr. Tarunraj Singh.

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