Forward flight of birds revisited. Part 1: aerodynamics and performance

This paper is the first part of the two-part exposition, addressing performance and dynamic stability of birds. The aerodynamic model underlying the entire study is presented in this part. It exploits the simplicity of the lifting line approximation to furnish the forces and moments acting on a single wing in closed analytical forms. The accuracy of the model is corroborated by comparison with numerical simulations based on the vortex lattice method. Performance is studied both in tethered (as on a sting in a wind tunnel) and in free flights. Wing twist is identified as the main parameter affecting the flight performance—at high speeds, it improves efficiency, the rate of climb and the maximal level speed; at low speeds, it allows flying slower. It is demonstrated that, under most circumstances, the difference in performance between tethered and free flights is small.

2. Introduction

Aerodynamics and flight mechanics of flapping flight have drawn research attention since the beginning of the aviation era. The complexity of aerodynamic models involved progressively increased, until recent advances in computing power have made full Reynolds-averaged Navier-Stokes simulations within reach [1]. It seems, however, that in the race for fidelity, a few fundamental problems became buried under excessive details. Two of these problems, performance and short-term dynamic stability of birds in forward flight, are revisited in this study with the simplest aerodynamic model feasible. The performance is addressed in this part; the short term dynamic stability is addressed in part 2 [2]. The aerodynamic model, serving both parts, is constructed herein.

By ‘simplest model feasible’, we understand a model that can furnish the aerodynamic loads in a closed analytical form, and is accurate enough to capture their behaviour in flapping flight. Taking the cue from [3], we construct this model in the framework of the basic lifting line theory. The present model differs from that of [3] in allowing additional degrees of freedom for the wings' motion, in furnishing the forces acting on a single wing as well as their first moments, and in remarkable simplicity of its final expressions. Using the vortex lattice method to provide the reference, the model is shown in §§4.5 and 5.2 to be accurate enough for all the purposes of this two-part study. It intrinsically limits it, however, to those flyers that generate lift without flow separation—that is, sufficiently large birds and bats (but not insects).

A wing invariably twists during flapping. The effect of twist on flight performance was addressed in many papers (e.g. [1,3,4]), but the twist never received the credit of being the most important parameter affecting the performance. It could have happened because of the historical definition of the propulsion efficiency: the ratio of the average power made good (the product of force produced by the wings in the direction of flight and airspeed) and the average power spent (see for example [4,5]). This definition ignores the double role played by flapping wings in flight. Stopping the wings makes the force generated by them in the direction of flight negative—in fact, this force becomes drag, parasite and induced combined. In comparison, it is commonly accepted that stopping the propeller of an aeroplane makes its thrust vanish—the drag of the aeroplane's wings is considered an inseparable part of the aeroplane's total drag. As propulsion efficiency of a propeller is defined irrespective of the aerodynamic characteristics of the aeroplane it propels, so the propulsion efficiency of flapping wings should be defined irrespective of the aerodynamic characteristics of the bird in non-flapping flight. Propulsion efficiency and its dependence on twist are addressed in §5.1.

A common measure of performance of a fixed wing aeroplane is the specific excess power [6]—essentially, it is the maximal sustained rate of climb at a given airspeed. We did not see the use of this measure in relation to flight performance of birds. Specific excess power of an aeroplane is limited by the engine power throughout the flight envelope. Specific excess power of a bird is also limited by the maximal power the bird can generate, but only at high speeds. Increasing the power at low speeds stalls the wings, putting an additional limit on performance. Specific excess power of a bird and its dependence on twist are addressed in §5.4.

Bird's body invariably pitches and heaves during flapping. Effects of the body motion on performance are addressed in §§6.1–6.3.

3. Kinematics

Consider a simplified symmetric bird in symmetric flight with constant velocity v. The mass of the bird is m; the density of the air in which it flies is ρ; acceleration of gravity is g; the length of a single wing (the semi-span) is s; its area is S; the aspect ratio of the two wings is A=2s²/S. s, v, vs, s/v, v/s, ρsv², ρs²v², ρSv², ρSv²s and ρSv³ will serve as convenient units of length, velocity, circulation, time, frequency, force per unit span, moment per unit span, force, moment and power, respectively. Note that although S is half the quantity commonly used as the wing area, the units of force and power are standard. Use of dimensionless quantities is implicitly understood hereafter. Should a dimensional quantity (other than ρ, g, m, s, S and v) be required, it will be marked by an asterisk. A list of nomenclature can be found in table 1.

Each wing is allowed to flap, sweep fore and aft, twist, heave and pitch. It is assumed that the wing twists in such a way that its sections do not deform and remain parallel to each other; moreover, the twist axis crosses all sections at their respective quarter-chord points and remains straight at all times. The sweep angle of the twist axis is λ (positive aft), flapping angle is ϕ (positive down); the twist angle is α_g (positive for leading edge up); pitch angle relative to the average flight path is τ (positive for nose up); vertical translation of the twist axis is h (figure 1). It is assumed that the twist varies linearly along the span, with

$${\alpha }_{\mathrm{g}}={\alpha }_{\mathrm{g}0}+{\alpha }_{g1}y,$$3.1

where y∈(0,1) is the spanwise coordinate; the description here pertains to the right wing. The twist can be active (through muscle contraction) or passive (through aerodynamic twisting moment); no attempt is made to model its intricate details.

Two right-handed Cartesian reference frames will be used. Frame L is rigidly connected to a local chord; its y-axis coincides with the twist axis of the right wing; its x-axis points backwards parallel to the chord; its z-axis points upwards and its origin rests in the symmetry plane of the bird (where the two wings meet). Frame C follows the bird with constant velocity (v) along a straight path; its x-axis points backwards along that path; its y-axis points right and its z-axis points upwards, through the origin of L. Unit vectors along the axes of L are ${\mathbf{e}}_x^{\mathrm{L}}$, ${\mathbf{e}}_y^{\mathrm{L}}$, ${\mathbf{e}}_z^{\mathrm{L}}$; unit vectors along the axes of C are ${\mathbf{e}}_x={\mathbf{e}}_x^{\mathrm{C}}$, ${\mathbf{e}}_y={\mathbf{e}}_y^{\mathrm{C}}$, ${\mathbf{e}}_z={\mathbf{e}}_z^{\mathrm{C}}$. Heave is manifested in time dependence of the distance h between the origins of L and C. Rotation from C to L is a series of four Euler's rotations: about the y-axis through angle τ; about the new z-axis through angle −λ; about the new x-axis through angle −ϕ and about the new y-axis through angle α_g. A comparable approach can be found in [5].

Explicit expressions for the components of ${\mathbf{e}}_x^{\mathrm{L}}$, ${\mathbf{e}}_y^{\mathrm{L}}$ and ${\mathbf{e}}_z^{\mathrm{L}}$ in C are lengthy, and hence are not written here for the general case. In the particular case where all angles are small when compared with unity

$$\begin{array}{rl}{\mathbf{e}}_x^{\mathrm{L}}& ={\mathbf{e}}_x-\mathrm{\lambda }{\mathbf{e}}_y-({\alpha }_{\mathrm{g}}+\tau ){\mathbf{e}}_z+\cdots ,\end{array}$$3.2

$$\begin{array}{rl}{\mathbf{e}}_y^{\mathrm{L}}& =\mathrm{\lambda }{\mathbf{e}}_x+{\mathbf{e}}_y-\varphi {\mathbf{e}}_z+\cdots \end{array}$$3.3

$$\begin{array}{rl}\text{and}{\mathbf{e}}_z^{\mathrm{L}}& =({\alpha }_{\mathrm{g}}+\tau ){\mathbf{e}}_x+\varphi {\mathbf{e}}_y+{\mathbf{e}}_z+\cdots ,\end{array}$$3.4

where the ellipses stand for the higher-order terms with respect to angles and their time derivatives. Concurrently, the velocity of a point on the wing relative to C is

$${\mathbf{v}}^{\mathrm{C}}=\dot{\mathrm{\lambda }}y{\mathbf{e}}_x-\dot{\mathrm{\lambda }}x{\mathbf{e}}_y+(\dot{\mathbf{h}}-\dot{\varphi }y-({\dot{\alpha }}_{\mathrm{g}}+\dot{\tau })x){\mathbf{e}}_z+\cdots ,$$3.5

where x and y are (by interpretation) the distances from the twist and flapping axes respectively, and an over-dot stands for derivative with respect to (dimensionless) time.

4. Aerodynamics

4.1 Assumptions

It is assumed that A⁻¹, α_g, τ, $\dot{h}$, ϕ, $\dot{\varphi }$, λ and $\dot{\mathrm{\lambda }}$ are small when compared with unity. The first assumption underlies the lifting line theory; the remaining assumptions underlie linearization. For the sake of simplicity of the following discussion, it is assumed that α_g, τ, $\dot{h}$ and $\dot{\varphi }$ are of comparable magnitudes, say Δ. The magnitude of λ is assumed not to exceed A⁻¹ and ϕ; $\dot{\mathrm{\lambda }}$ is assumed to be a second-order quantity with respect to ϕ, Δ and their products.

It is postulated that a vortical wake exists past the wing, starting at the trailing edge and extending to infinity. It is assumed that the vorticity is constant along that portion of the wake adjacent to the trailing edge that affects the flow over the wing. This assumption implies that the flapping frequency ω is sufficiently small; it is plausible—in fact, it will be shown by example—that ω can be of the order of unity. The aerodynamic model developed herein is coherent only in the leading order with respect to ϕ, Δ, and their products with ω.

It is assumed that the wing has an elliptical plan-form with chord length prescribed by

$$c=c_0\sqrt{1-y^2}=\frac{8}{\pi A}\sqrt{1-y^2};$$4.1

c₀=8/πA, because S=s²c₀π/4 and A=2s²/S by definition. Exploiting the symmetry of the problem, the range of y in all subsequent equations is extended to (−1, 1); negative values corresponding to the left wing.

4.2 Fundamentals

As already mentioned in §2, the aerodynamic model for this study is based on the classical (quasi-steady) lifting line theory ([7, p. 586]). In brief, this theory associates the lift of a wing section, represented by the right-hand side of the following equation, with the lift of an equivalent vortex, represented by the left-hand side

$$\mathit{\Gamma }=\frac{1}{2}ca(\alpha -{\alpha }_{\mathrm{i}})\mathrm{for}\hspace{0.17em}\mathrm{each}\hspace{0.17em}y\in (-1,1).$$4.2

Γ is the circulation of that vortex, a=2π is the lift-slope coefficient of the wing section, α is the effective angle of attack the wing section relative to unperturbed fluid (that can be considered known), and α_i is the angle of attack induced by the wake (figure 2). The closure of (4.2) is obtained by relating α_iwith Γ by Biot–Savart's law ([7], p. 94) which transforms (4.2) into an integro-differential equation for Γ. Once solved, the aerodynamic loads follow by quadratures.

The non-intuitive elements in this paradigm are the angles α and α_i. In the lifting-line limit, the wing reduces to a vortex in the y-z plane whereas the wake reduces to the sheet of vortices starting at the wing vortex and extending to infinity in the positive x-direction [8]. Coherent with this model, α is approximated by its leading-order term (with respect to the aspect ratio), $\underset{x\to 0}{lim}({\mathbf{e}}_z^{\mathrm{L}}\cdot ({\mathbf{e}}_x-{\mathbf{v}}^{\mathrm{C}})/|{\mathbf{e}}_x-{\mathbf{v}}^{\mathrm{C}}|)$. For a non-cambered wing, it yields

$$\begin{array}{rl}& \alpha =\tau +{\alpha }_{\mathrm{g}}+\dot{\varphi }|y|-\dot{h}+\cdots ={\alpha }_0+{\alpha }_1|y|+\cdots ,\end{array}$$4.3

$$\begin{array}{rl}& {\alpha }_0={\alpha }_{\mathrm{g}0}+\tau -\dot{h}\end{array}$$4.4

$$\begin{array}{rl}\mathrm{and}& {\alpha }_1={\alpha }_{\mathrm{g}1}+\dot{\varphi },\end{array}$$4.5

by (3.1), (3.4) and (3.5); it can be extended ad hoc for a cambered wing by defining α_g as the angle between the x-axis of C and the zero-lift line (rather than the chord) of the respective section. α₀ will be recognized as the spanwise uniform constituent of the angle of attack—it is associated with pitch and heave of the body and twist at the shoulder; α₁ is the gradient of the angle of attack along the span—it is associated with the flapping rate and with spanwise-variable twist.

α_i is approximated by its leading-order term with respect to the aspect ratio as well—that is, the normal-to-the-wing (reduced) velocity component in the y-zplane induced by the wake [8]. Because ϕ is small when compared with unity, and the wake vorticity is supposedly constant along that portion of the wake that affects the flow about the wing (see §4.1),

$${\alpha }_{\mathrm{i}}=\frac{1}{4\pi }--{\int }_{-1}^1\frac{\mathrm{\partial }{\mathit{\Gamma }}^{\prime }}{\mathrm{\partial }{y}^{\prime }}\frac{\mathrm{d}{y}^{\prime }}{y-y^{\prime }}+\cdots .$$4.6

A combination of (4.2) and (4.6) leads to the well-known integrodifferential equation for Γ,

$$\mathit{\Gamma }+\frac{ac}{8\pi }--{\int }_{-1}^1\frac{\mathrm{\partial }{\mathit{\Gamma }}^{\prime }}{\mathrm{\partial }{y}^{\prime }}\frac{\mathrm{d}{y}^{\prime }}{y-y^{\prime }}=\frac{1}{2}ac\alpha \mathrm{for}\hspace{0.17em}\mathrm{each}\hspace{0.17em}y\in (-1,1),$$4.7

in which α is given by (4.3), c is given by (4.1), and, in general, a=2π. Its relevant solution is

$$\mathit{\Gamma }=4\sum _{n=1,3,\dots }^{\mathrm{\infty }}\hspace{0.17em}{a}_n\sin n\theta ,$$4.8

where

$$\begin{array}{rl}\theta & ={\cos }^{-1}(-y),\end{array}$$4.9

$$\begin{array}{rl}{a}_{2n-1}& =A_{2n-1}\left({\alpha }_0{\delta }_{n1}-\frac{4}{\pi }{\alpha }_1{I}_{1,2n-1}\right),\end{array}$$4.9

δ_nm is Kronecker's delta, $I_{mn}={\int }_{\pi /2}^{\pi }\sin n\theta \hspace{0.17em}\sin m\theta \hspace{0.17em}\cos \theta \mathrm{d}\theta$ are standard integrals, and

$$A_n=\frac{a}{\pi A+na}=\frac{2}{A+2n}.$$4.11

Details can be found in appendix A. Explicit expression for α_i,

$${\alpha }_{\mathrm{i}}=\frac{1}{4\pi vs}--{\int }_0^{\pi }\frac{\mathrm{\partial }{\mathit{\Gamma }}^{\prime }}{\mathrm{\partial }{\theta }^{\prime }}\frac{\mathrm{d}{\theta }^{\prime }}{\cos {\theta }^{\prime }-\cos \theta }=\sum _{n=1,3,\dots }^{\mathrm{\infty }}\hspace{0.17em}\hspace{0.17em}n{a}_n\frac{\sin n\theta }{\sin \theta },$$4.12

follows (4.6) by (4.8) and (A 2).

4.3 Forces and moments