U-shaped power curve
A variety of mathematical models may be used to estimate the effects of flight speed upon components of Paero (Norberg, 1990). Models that are mostly widely employed are those of Pennycuick (Pennycuick, 1975; Pennycuick, 1989) and Rayner (Rayner, 1979a) for forward flight and, for hovering, those of Rayner (Rayner, 1979b) and Ellington (Ellington, 1984). Regardless of which model is used, the general prediction that always emerges is that Paero should vary with flight speed according to a U-shaped curve, with greater Paero required during hovering and fast flight and less required during flight at intermediate speeds (Norberg, 1990) (Fig. 2A). As a function of forward flight velocity, the cost of producing lift, Pind, decreases, while power needed to overcome drag on the wings and body, Ppro and Ppar, respectively, increases.
The U-shaped curve for Paero features a characteristic minimum power speed (Vmp) and a maximum range speed (Vmr; Fig. 2A). These characteristic speeds represent one of the most obvious ways in which the biomechanics of flight may be integrated with behavioral ecology. Often, a starting premise for ecological studies of flight is that birds should select Vmp for aerial foraging or searching and select Vmr for long-distance flight such as migration, although specific predictions change when optimal foraging factors such as rate of energy intake or prey-delivery rates are incorporated into the models (Hedenström and Alerstam, 1995; Houston, 2006).
Consistent with the hypothesis that the pectoralis is the primary muscle supplying the mechanical power output required for flight, measures of Pmusin vivo (Tobalske et al., 2003) (Fig. 2B) and in vitro (Askew and Ellerby, 2007) vary as U-shaped curves in the same manner as Paero. The shape of a Pmus curve is affected by the morphology and flight style of a given species, and this means that there is inter-specific variation in Vmp and Vmr. Some models of Paero may be refined to take into account details of wing shape, wing kinematics and intermittent flight behavior (Rayner, 1979a; Rayner, 1979b; Ellington, 1984; Rayner, 1985). Thus, efforts to better understand both the biomechanics and ecology of flight will benefit from revision of these models as additional empirical evidence emerges.
Small differences in efficiency (i.e. Pmus/Pmet) have the potential to dramatically affect the shape of the Pmet power curve relative to that of Pmus and Paero (Thomas and Hedenström, 1998; Rayner, 1999). Since Pmet is the rate of energy input by a bird, the shape of the Pmet curve, rather than that of Pmus or Paero, is what is ultimately of importance in governing the Vmp and Vmr of interest to a bird. For many years, it appeared that the curve for Pmet was flat at intermediate flight speeds, which suggested that efficiency is lowest in the range of preferred flight speeds (reviewed in Ellington, 1991). Recently, efforts to measure Pmet using a variety of techniques including double-labelled water, heat transfer and gas respirometry, reveal that the curve for Pmet is U-shaped much as that for Pmus and Paero, with muscular efficiency in the range of 20% (Ward et al., 2001; Ward et al., 2004; Engel et al., 2006; Bundle et al., 2007) (Fig. 2C). Comparison of the curves for Pmet and Pmus in cockatiels Nymphicus hollandicus, measured in two different studies, suggests that efficiency is not constant across speeds and that Vmp and Vmr are both faster when measured using Pmet compared with Pmus (Tobalske et al., 2003; Bundle et al., 2007) (Fig. 2). To further clarify this issue, it is vital to measure Pmet and Pmus under similar experimental conditions due to potential differences in Paero when a mask and respirometery cabling is added to an animal (Bundle et al., 2007).
The major difference among models of Paero is the method used to estimate Pind, although different approaches are also employed to estimate Ppro and Ppar. In a relatively simple model that is widely used by ecologists, in part because it is available as a computer program (Pennycuick, 1975; Pennycuick, 1989), Pind is estimated using a steady-state momentum-jet model. This model treats the wings as fixed-length propellers rotating and translating at a steady rate, as in a helicopter, even for the gliding flight of birds. Propellers do not fully represent the complexity of the motions and morphing ability of bird wings (Bilo, 1971; Bilo, 1972; Warrick et al., 2005; Lentink et al., 2007). In contrast, alternative models (Rayner, 1979a; Rayner, 1979b; Ellington, 1984) use vortex theory, which is capable of incorporating unsteady motion and long-axis rotation of the wings (e.g. Dickinson et al., 1999).
U-shaped curves of power as a function of flight speed in birds. (A) Estimated mechanical power output required for flight in a European kestrel Falco tinnunculu (from Rayner, 1999). Paero, total aerodynamic power, Pind, induced power, Ppar, parasite power and Ppro, profile power, Vmp, velocity for minimum power, Vmr, velocity for maximum range. (B) In vivo mechanical power output from wind-tunnel flight across flight speeds as measured using strain gauges, sonomicrometry and wing and body kinematics in dove Zenaida macroura, cockatiel Nymphicus hollandicus, magpie Pica hudsonica (from Tobalske et al., 2003). (C) Oxygen consumption, an index of metabolic power output, measured in cockatiels over a range of flight speeds using gas respirometry (from Bundle et al., 2007).
When a wing produces lift, there is a net circulation of air about the wing that represents a bound vortex on the wing (Rayner, 1979a; Rayner, 1979b; Ellington, 1984). The term `bound' in this instance means attached or close to the wing, and is a definition independent of the flexed-wing bound posture used in intermittent flight. For real-world wings of finite span, the bound vortex is shed into the wake as a `wake' vortex, and the circulation in the wake vortex is equal in magnitude but opposite in sign to that of the bound vortex. Circulation varies with translational and long-axis rotational velocity, angle of attack and camber of the wing. Lift, in turn, is proportional to circulation as well as wing span, translational velocity and air density. One simple way to think of the relationship between wake vortices and lift is that, for a given need for lift, as surface area surrounded by the wake vortices increases, the velocity that the wings induce into the wake and the cost of producing lift, Pind, both decrease.
During the 1980s there was an earnest attempt to test vortex theory for flying birds using particle image velocimetry (PIV). Although the geometry of the vortices shed into the wake during slow flight matched expectations, the researchers were frustrated by measurements of momentum in the wake that were insufficient to support the weight of the bird (Spedding et al., 1984). This `momentum deficit paradox' was resolved using modern DPIV (Spedding et al., 2003), which offers finer resolution of flow patterns in the wake. Significant new observations about wake structure will undoubtedly improve models of Pind.
For example, based on wake samples in the European kestrel Falco tinnunculus engaged in moderate-speed forward flight (Spedding, 1987), it was formerly thought that birds varied lift primarily using wing flexion, and not by changing circulation in the bound vortex on their wings via changes in wing velocity, camber or angle of attack (Rayner, 1988). Instantaneous changes in circulation on the wings initiate the shedding into the wake of `cross-stream' vortices that are parallel to long-axis of the wings. These cross-stream vortices traverse the wake, they reveal a reduction in the effective area for lift production that is swept by the wings, and this reduction in effective area is predicted to increase the cost of producing lift, Pind (Rayner, 1988). Assuming a lack of significant cross-stream vortices, the kinematics of faster flight in birds such as kestrels or pigeons (Columba livia) emerge as an optimal pattern of wing motion (Rayner, 1999) (Fig. 3A). Modern DPIV reveals that cross-stream vortices are typical of forward flight (Spedding et al., 2003; Hedenström et al., 2006) (Fig. 4), and this should serve to revise models of Pind during fast flight.
Wing kinematics differ depending upon a bird's wing design and flight speed. (A) Birds with pointed, high-aspect ratio wings such as the pigeon Columba livia transition from tip-reversal upstrokes during slow flight to feathered upstrokes at intermediate speeds and a swept-wing upstroke during fast flight. (B) Birds with rounded, low-aspect ratio wings such as the black-billed magpie Pica hudsonica use a flexed upstroke at all flight speeds. Shown are wingtip (filled circles) and wrist (open circles) paths in dorsal and lateral view (from Tobalske and Dial, 1996).
Likewise, DPIV has recently revised our understanding of the mechanics of hovering (Warrick et al., 2005). Hummingbirds are the only birds that can sustain hovering in still air, and formerly it was thought that they supported their weight during hovering using symmetrical down and upstrokes with equal Pind during each half of the wingbeat. This assumption lead to a proportionally lower estimate for Pind compared with all other bird species, in which it appears that only the downstroke helps to support weight during hovering and slow flight (Rayner, 1979a). DPIV instead reveals that approximately 75% of the weight support during a wingbeat is provided by downstroke, and 25% is provided by upstroke, in hovering rufous hummingbirds Selasphorus rufus (Warrick et al., 2005).
Representations of vortex wakes shed from the wings of a thrush nightingale Luscinia luscinia at slow (A), medium (B) and fast (C) flight speeds in a wind tunnel, measured using digital particle image velocimetry [DPIV (from Spedding et al., 2003)]. Red and blue indicate the wake from upstroke and downstroke, respectively. Both phases of the wingbeat are aerodynamically active at each speed, and there are prominent cross-stream vortices apparent at the ends of half-strokes during slower flight (A,B) and throughout the wingbeat cycle during faster flight (C).
Reasonable measurements of Ppro are largely lacking for birds engaged in flapping flight, and this highlights a clear need for new research. Models presently either assume that profile drag is constant during intermediate flight speeds (Pennycuick, 1975; Pennycuick, 1989) or apply a coefficient of drag for the wing that is obtained from fixed-wing measurements and modeled according to blade-element theory, which treats the wing as set of fixed-width strips each moving at their own velocity, due to the flight velocity of the bird as well as the angular velocity of the wing (Rayner, 1979b). Unfortunately, drag on fixed wings (Withers, 1981; Lentink et al., 2007) or during gliding in live birds (Pennycuick et al., 1992) is probably quite different from unsteady drag forces operating on the flapping wings during slow flight (Spedding, 1993; Dickinson, 1996). Promising methods for more accurate models of Ppro during flapping include measuring force using transducers at the base of mounted, revolving wings (Usherwood and Ellington, 2002) or robotic flapping wings (Sane and Dickinson, 2001), perhaps coupled with computational fluid dynamics (Wang et al., 2004).
Caution is always merited when interpreting the measurements from revolving wings because propeller motion does not fully emulate the complex motion of the bird wing. Bearing this in mind, it may be that drag is higher during rotational motion of the wings compared with gliding. Maximal drag coefficients for the wings of galliform birds (Phasianidae) during rotation (Usherwood and Ellington, 2002) are approximately double the value measured for translating wings (Drovetski, 1996).
To date, dynamically scaled robotic models have only been developed for insects (Willmott et al., 1997; Sane and Dickinson, 2001), and these models have wings designed as flat plates that do not morph like bird wings (Bilo, 1971; Bilo, 1972; Warrick et al., 2005). Ideally, efforts to physically model Ppro will incorporate detailed 3D kinematics of the wing (Askew et al., 2001; Hedrick et al., 2002; Tobalske et al., 2007) to program robotic motion and use materials for the robotic wing that morph in a realistic manner.
As for Ppro, estimates of Ppar are subject to considerable uncertainty because drag coefficients obtained from isolated bird bodies vary over 400%, from ∼0.1 in varnished, footless, starling carcasses (Maybury, 2000) to ∼0.4 in frozen waterfowl (Pennycuick et al., 1988). One might expect values from live birds to be lower than frozen carcasses, but a parasite drag coefficient of ∼0.37 was measured from dive rates in passerines during migration (Hedenström and Liechti, 2001). Coefficients from 0.24–0.34 may be representative of live starlings in flight, and drag coefficients decrease as a function of forward flight speed (Maybury, 2000). There is some debate over the manner in which body area scales with body mass among species, and frontal projected area is necessary for a computation of coefficient of drag (Norberg, 1990). Hedenström and Rosén suggest that the frontal area of the body scales differently in passerines and non-passerines (Hedenström and Rosén, 2003), whereas Nudds and Rayner argue that scaling is similar between the two groups, and that reported differences in other studies are caused by comparing live birds with frozen specimens (Nudds and Rayner, 2006).
To further understanding of the aerodynamics of the bird body, it will be useful to take a broader view of the body to include the tail (Thomas, 1993). Treating the body as a `parasite' upon the wings is a leftover from early aerodynamics research and is misleading, because the body is capable of producing lift even with the wings completely folded, as during intermittent bounds, which are flexed-wing pauses in between flapping phases (Csicsáky, 1977; Tobalske et al., 1999). The tail functions to reduce parasite drag (Maybury and Rayner, 2001); it contributes to the production of lift both when the wings are not present on a carcass (Maybury, 2000; Maybury et al., 2001) as well as during flight in live birds (Usherwood et al., 2005). Incorporating body lift (and, by extension, tail lift) into a model of Paero reduces the estimated power required for relatively fast flight in flap–bounding birds (Rayner, 1985; Tobalske et al., 1999).
A model that is useful for describing the aerodynamics of the tail considers the tail as delta wing (Thomas, 1993). This model indicates that the tail morphology that produces the optimum lift-to-drag ratio is slightly forked when folded and triangular in shape when fanned, and that the area of the tail in front of the maximum span contributes to lift and drag, whereas the area behind the maximum span contributes only to drag. These properties of the tail serve as a foundation for testing, with vigorously debated conclusions, whether the evolution of ornamentation, such as an elongated tail or streamers on a forked tail, represent a handicap that resulted from sexual selection or an aerodynamic benefit for activities such as maneuvering (Thomas, 1993; Møller et al., 1995; Park et al., 2000; Evans, 2004). Aspects of the model do not appear to be well supported by the use of the tail during flight in barn swallows Hirundo rustica (Evans et al., 2002), so new efforts are needed to modify the delta-wing model or develop alternative models.
Kinematics and upstroke aerodynamics
Unlike some forms of terrestrial locomotion in which gait selection may be identified using discrete changes in duty factor (the proportion of time a limb is in contact with the substrate, producing force) and patterns of sequential limb motion (Alexander, 1989), wing kinematics and associated aerodynamics, vary in a continuous manner with flight speed in birds.
Although downstroke kinematics are relatively invariant, upstroke kinematics differ among species and according to flight speed (Brown, 1963; Scholey, 1983; Tobalske, 2000; Park et al., 2001; Hedrick et al., 2002). Birds with wings that are relatively pointed, or of high-aspect ratio (long and thin), transition among flight speeds using tip-reversal upstrokes at slow speeds, feathered upstrokes at intermediate speeds, and swept-wing upstrokes at fast flight speeds (Fig. 3B). Birds that have rounded distal wings or wings of low aspect ratio (short and broad) tend to flex their wings regardless of flight speed. Some exceptions to this pattern exist. For example, galliform birds with rounded wings use a tip-reversal upstroke during take-off (Brown, 1963; Tobalske and Dial, 2000), and birds with rounded wings such as the black-billed magpie Pica hudsonica will alter upstroke postures according to acceleration and deceleration (Tobalske and Dial, 1996). Regardless of wing shape, the span ratio (mid-upstroke span divided by mid-downstroke span) generally decreases as a function of flight speed in birds, although it may increase with increasing speed in some passerines (Tobalske and Dial, 1996; Tobalske et al., 1999; Rosén et al., 2004; Tobalske et al., 2007).
Formerly, it was thought that there were two wingbeat gaits in birds because early PIV experiments revealed one of two patterns. One vortex ring was shed per downstroke during slow flight (Spedding et al., 1984), and the upstroke appeared aerodynamically inactive. This was identified as a `vortex-ring' gait (Rayner, 1988; Rayner, 1999). During faster flight (Spedding, 1987), tip-vortices were shed into the wake during the entire wingbeat, indicating a `continuous-vortex' gait (Rayner, 1988; Rayner, 1999). Because the wake area for a continuous-vortex wake would be greater than for a vortex ring, Pind was predicted to be lower for the continous-vortex gait.
Several problems are, nevertheless, apparent with a simple two-gait scheme for classifying avian flight (Tobalske, 2000), and new data reveal that continuous, rather than discrete, variation is characteristic of wing kinematics and aerodynamics (Spedding et al., 2003; Rosén et al., 2004;
Steady flight, unaccelerated flight, or equilibrium flight is a special case in flight dynamics where the aircraft's linear and angular velocity are constant in a body-fixed reference frame. Basic aircraft maneuvers such as level flight, climbs and descents, and coordinated turns can be modeled as steady flight maneuvers. Typical aircraft flight consists of a series of steady flight maneuvers connected by brief, accelerated transitions. Because of this, primary applications of steady flight models include aircraft design, assessment of aircraft performance, flight planning, and using steady flight states as the equilibrium conditions around which flight dynamics equations are expanded.
- Origin - airplane center of gravity
- xw axis - positive in the direction of the velocity vector of the aircraft relative to the air
- zw axis - perpendicular to the xw axis, in the plane of symmetry of the aircraft, positive below the aircraft
- yw axis - perpendicular to the xw,zw-plane, positive determined by the right hand rule (generally, positive to the right)
The Euler angles linking these reference frames are:
- Earth frame to body frame: yaw angle ψ, pitch angle θ, and roll angle φ
- Earth frame to wind frame: heading angle σ, flight-path angle γ, and bank angle μ
- Wind frame to body frame: angle of sideslip β, angle of attack α (in this transformation, the angle analogous to φ and μ is always zero)
Force balance and the steady flight equations
The forces acting on an aircraft in flight are the weight, aerodynamic force, and thrust. The weight is easiest to express in the Earth frame, where it has magnitude W and is in the +zE direction, towards the center of the Earth. The weight is assumed to be constant over time and constant with altitude.
Expressing the aerodynamic force in the wind frame, it has a drag component with magnitude D opposite the velocity vector in the −xw direction, a side force component with magnitude C in the +yw direction, and a lift component with magnitude L in the −zw direction.
In general, the thrust can have components along each body frame axis. For fixed wing aircraft with engines or propellers fixed relative to the fuselage, thrust is usually closely aligned with the +xb direction. Other types of aircraft, such as rockets and airplanes that use thrust vectoring, can have significant components of thrust along the other body frame axes. In this article, aircraft are assumed to have thrust with magnitude T and fixed direction +xb.
Steady flight is defined as flight where the aircraft's linear and angular velocity vectors are constant in a body-fixed reference frame such as the body frame or wind frame. In the Earth frame, the velocity may not be constant since the airplane may be turning, in which case the airplane has a centripetal acceleration (Vcos(γ))2/R in the xE-yE plane, where V is the magnitude of the true airspeed and R is the turn radius.
This equilibrium can be expressed along a variety of axes in a variety of reference frames. The traditional steady flight equations derive from expressing this force balance along three axes: the xw-axis, the radial direction of the aircraft's turn in the xE-yE plane, and the axis perpendicular to xw in the xw-zE plane.
where g is the standard acceleration due to gravity.
These equations can be simplified with several assumptions that are typical of simple, fixed-wing flight. First, assume that the sideslip β is zero, or coordinated flight. Second, assume the side force C is zero. Third, assume that the angle of attack α is small enough that cos(α)≈1 and sin(α)≈α, which is typical since airplanes stall at high angles of attack. Similarly, assume that the flight-path angle γ is small enough that cos(γ)≈1 and sin(γ)≈γ, or equivalently that climbs and descents are at small angles relative to horizontal. Finally, assume that thrust is much smaller than lift, T≪L. Under these assumptions, the equations above simplify to
These equations show that the thrust must be sufficiently large to cancel drag and the longitudinal component of weight. They also show that the lift must be sufficiently large to support the aircraft weight and accelerate the aircraft through turns.
Dividing the second equation by the third equation and solving for R shows that the turn radius can be written in terms of the true airspeed and the bank angle,
The constant angular velocity in the body frame leads to a balance of moments, as well. Most notably, the pitching moment being zero puts a constraint on the longitudinal motion of the aircraft that can be used to determine the elevator control input.
Steady flight maneuvers
The most general maneuver described by the steady flight equations above is a steady climbing or descending coordinated turn. The trajectory the aircraft flies during this maneuver is a helix with zE as its axis and a circular projection on the xE-yE plane. Other steady flight maneuvers are special cases of this helical trajectory.
- Steady longitudinal climbs or descents (without turning): bank angle μ=0
- Steady level turns: flight-path angle γ=0
- Steady level longitudinal flight, also known as "flying straight and level": bank angle μ=0 and flight-path angle γ=0
- Steady gliding descents, whether turning or longitudinal: thrust T=0
The definition of steady flight also allows for other maneuvers that are steady only instantaneously if the control inputs are held constant. These include the steady roll, where there is a constant and non-zero roll rate, and the steady pull up, where there is a constant but non-zero pitch rate.