## SUMMARY

To seek the simplest efficient flapping wing motions and understand their
relation to steady flight, a two-stroke model in the quasi-steady limit was
analyzed. It was found that a family of two-stroke flapping motions have
aerodynamic efficiency close to, but slightly lower than, the optimal steady
flight. These two-stroke motions share two common features: the downstroke is
a gliding motion and the upstroke has an angle of attack close to the optimal
of the steady flight of the same wing. With the reduced number of parameters,
the aerodynamic cost function in the parameter space can be visualized. This
was examined for wings of different lift and drag characteristics at Reynolds
numbers between 10^{2} and 10^{6}. The iso-surfaces of the
cost function have a tube-like structure, implying that the solution is
insensitive to a specific direction in the parameter space. Related questions
in insect flight that motivated this work are discussed.

## Introduction

Quantitative studies have made much progress in revealing various aerodynamic mechanisms for force generation in flapping flight, as extensively reviewed (Weis-Fogh and Jensen, 1956; Ellingon, 1984; Dickinson, 1996; Sane, 2003; Lehmann, 2004; Wang, 2005). Less clear is the relative cost of various wing motions that are capable of generating the same averaged force. For example, for a given wing, how does the aerodynamic cost of flapping flight compare to that of a fixed wing flight? A necessary first step in addressing the question of relative efficiency is to define a fair and relevant measure for cross-comparing different wing motions. With the model of aerodynamic forces and actuators, we can, in principle, determine efficient wing motions either in real animals or model systems. Suppose we succeed in doing so, the solutions may still be non-intuitive due to the fact that a typical wing motion is described by a large number of parameters. Moreover, some features in the predicted motions are specific to the model rather than the original system. The main purpose of this paper is to seek some common features of efficient flapping motions in a minimal model, as a step toward understanding the more complex ones.

Hovering, as opposed to forward flight, is a natural candidate for
comparing the efficiency of two flight strategies: flapping *vs* steady
flight. In hovering, a moving wing alone generates the required thrust without
the need for additional propulsions. Here, the word efficiency is used to mean
the inverse of the dimensionless cost. First, an energetic criterion is
defined, with which the relative aerodynamic cost of employing different wing
motions can be compared. Finding efficient wing motions requires an effective
method for reducing the parameter space without excluding all of the efficient
motions. One approach is to use the observed insect wing motions as a guide to
construct families of wing motions described by a set of physical parameters.
The idea is that the odds of finding efficient solutions among insects' or
birds' wing motions are likely to be higher than our random guesses. In the
reduced parameter space, we can identify with energy-minimizing flapping wing
kinematics for various insects, compare with observed motions, and recognize
common features (Berman and Wang,
2007). In this paper, we seek the simplest efficient flapping wing
motions that can be analyzed in detail. To this end, we consider a family of
up and down motions described by six parameters and calculate the aerodynamic
power in quasisteady limit. The parameter space was further reduced to
four-dimensions, based on the reasoning described below. An advantage of
working with the remaining four-parameter space is that it is possible to
visualize the parameter space. The sensitivity of the cost function can be
viewed with respect to the wing motion parameters, and in relation to
classical steady wing motion.

## A criterion for comparing the aerodynamic cost of a wing undergoing different hovering wing motions

For animals and airplanes, the total energetic cost is measured by the consumption of their respective fuels. At the limit where the conversion rate from chemical to mechanical energy is independent of the wing motion, the total cost is directly proportional to the mechanical work. The mechanical work done by a flapping wing includes aerodynamic and inertial components. The former is the work done to overcome fluid drag and the latter, work done to accelerate and decelerate the wing in a vacuum. The inertial cost can be calculated directly from the wing kinematics, and unless the elastic storage of the muscles is perfect, the net inertial cost is non-zero. The aerodynamic cost can be measured experimentally, calculated using direct numerical simulations, or estimated by quasi-steady force models.

The aerodynamic efficiency of transport of a classical airfoil in steady forward flight is determined by the lift:drag ratio, which is the inverse of the aerodynamic work required to transport a unit weight over a unit distance. A similar ratio can be defined for the efficiency of endurance, which is proportional to the inverse of the aerodynamic power required to support a unit weight. These ratios are often used to compare the relative efficiency of different airfoils: the higher the ratio, the more efficient the airfoil. For flapping motions, in addition to cross-comparing different wing shapes, it is of interest to investigate the relative efficiency of the same airfoil undergoing different wing motions, which is the focus of this work.

In the case of a hovering insect, the wing and the weight are given, and we
seek the wing motion that minimizes the mechanic power subject to the
constraint of the weight balance. Specifically, the aerodynamic cost of
endurance (*P*^{*}) is defined as the dimensionless aerodynamic
power to support a unit weight:
(1)
with the constraint:
(2)
where and
are instantaneous aerodynamic force
and wing translational velocity,
and
the instantaneous aerodynamic
torque and wing angular velocity, ** Mg** the weight, and

*T*the period.

*P*

^{*}is dimensionless, and the reference velocity

*U*

_{ref}= is constant for a specific wing of area

*A*and weight

**. The inertial cost can be added if we have a model of elastic storage of the muscles (Berman and Wang, 2007). For the piece-wise constant motions studied here, the inertial cost is zero except near the transition.**

*Mg*Note that in the case of steady wing motion
(Fig. 1B),
*P*^{*} has the familiar form:
*P*^{*}=*C*_{D}(α)/(α),
where *C*_{L}(α) and *C*_{D}(α) are
lift and drag coefficients, respectively. For general wing motions, this
simple ratio no longer holds, and thus maximizing the averaged lift:drag ratio
is not equivalent to minimizing the aerodynamic power. We further note that
*P*^{*} is proportional to the specific power, power per mass
supported (Ellington, 1984).
The main difference is that here the specific power is only compared among the
motions that generate the same force. This difference matters when the
aerodynamics force coefficients depend indirectly on the net force due to, for
example, the change of the Reynolds number of the flow required to generate
the specified force.

## Two-stroke wing motions and a model of quasi-steady forces

The steady forward flight motion is defined by the wing velocity,
*U*, and the angle of attack, α (stroke-I,
Fig. 1A). Similarly, the rotary
wing motion is defined by the angular velocity, Ω and α. The
simplest flapping motion is a back-and-forth motion, which can be viewed as a
rotary motion projected onto the diameter keeping the same angle of attack and
velocity (Fig. 1B). In the
quasi-steady limit considered here, these symmetrical back and forth motions
are equivalent to the rotary motion. Among them, the one that minimizes
*P*^{*} has α_{m} that minimizes
and
*U*_{m} that balances the weight at
.

The next simplest flapping motion (Stroke-II,
Fig. 1E) consists of two
constant motions arranged in a V-shape, and is defined by seven parameters:
the velocity (*U*_{d,u}), the angle of attack
(α_{d,u}), the angle of stroke path (β_{d,u})
during down (d) and up (u) strokes, and the fraction of a period spent on the
upstroke (ϵ). Six of them are independent if we further require that two
ends of V maintain the same altitude. A pair of mirror images of each V-shape
forms a figure-eight (Fig. 1F),
which is a hovering motion.

These two-stroke motions are piece-wise constant, thus the main unsteady
aerodynamic effect is the dynamic stall, during which the leading edge vortex
forms and remains attached while producing a force that is roughly constant at
a given angle (Francis and Cohen,
1933; Ellington,
1984; Dickinson and Götz,
1993; Ellington et al.,
1996; Wang, 2000;
Usherwood and Ellington,
2002). At this limit, the empirical formula to fit the lift-drag
at relative low Reynolds numbers (∼10^{2}) can be expressed as
(Wang, 2005):
(3)
(4)
where *C*_{L} and *C*_{D} are lift and drag
coefficients, respectively, α the angle of attack, α_{0l}
the angle at which lift is zero and α_{0d} the angle at which
the drag is minimal. α_{0l} and α_{0d} are
non-zero for asymmetrical airfoils. Similar form was also obtained to fit the
experimental measurements of lift–drag characteristics of helicopter
blades during dynamic stall at much higher Reynolds numbers of about
10^{6} (Leishman,
2000). The fit of this formula to airfoil data is shown in
Fig. 2. Note that
Eqn 3 differs from the classical
Joukowski's lift in its angle dependence, sin2α rather than sinα,
and it is to be applied to all angles of attack during dynamic stall.

## Efficient two-stroke wing motions

In this minimal model, the parameter space is six-dimensional, which is still difficult to visualize directly. To further reduce the number of parameters, we make two observations. The first is that the classical airfoil motion of a fixed or rotating wing uses only aerodynamic lift, but not drag, to support a weight (Wang, 2004). If the airfoil is reoriented such that the net force is vertical (Fig. 1C), as in gliding, the wing can support an additional weight by a factor of . Consider the ideal case in which the downstroke is a gliding motion and the upstroke returns instantaneously, consuming no energy, to support a specified weight, and . This gliding motion is more efficient and is used as the downstroke of the flapping motions studied below.

An upstroke must return in a finite time and it costs energy. At first
sight, the least costly upstroke seems to be the vertical upstroke at zero
angle of attack (Fig. 1D),
because it moves along the shortest path and the wing experiences the least
drag. However, this is not the case (Fig.
3). To find a more efficient upstroke, we note the angle
dependence of the aerodynamic force at small angles. As α deviates from
zero, to the leading order, the lift increases as α, as given by the
Kutta–Joukowski theory, while the drag increases as α^{2}.
At small α,α
^{2}α,
thus an upstroke at small angle of attack generates a lift at a relatively
small cost, which can be advantageous compared to that with α=0, which
generates no lift. These lift-generating strokes replace the vertical
upstroke. The two-stroke motion is now described by four parameters,
(α_{u}, β_{u}, ϵ, α_{d}), and
the corresponding *P*^{*} is given by:
(5)
where
(6)
(7)
(8)
We minimize *P*^{*} with respect to (ϵ,β
_{u}, α_{u}) for each downstroke parameterized byα
_{d} in Matlab.

The cost to transit between the two piece-wise constant strokes is
neglected in the calculation of *P*^{*}. Its contribution is
small compared to the rest of the stroke in the large stroke-amplitude limit.
More interestingly, the wing pitch reversal does not necessarily require
additional power (Bergou et al.,
2007; Berman and Wang,
2007), which is discussed later.

Fig. 3A compares
1/*P*^{*} among three kinds of wing motions: back-and-forth
motion with a constant angle of attack (α_{d}) two-stroke,
composed of a gliding downstroke with the same angle of attack
(α_{d}) followed by a vertical upstroke at zero angle of attack,
and two-stroke composed of the same gliding stroke followed by an optimized
upstroke. Two representative Reynolds numbers (*Re*) are shown:
10^{3}, close to that of an insect, and 3×10^{5}, close
to that of a low-speed airplane. The maximum lift:drag ratios are 4.7 and 60,
respectively. In both cases, there are multiple asymmetrical down-and-up
flapping motions (Fig. 3B),
whose *P*^{*} are close to minimal, but slightly higher than
the minimum of
of
the steady wing motion.

By definition, the first derivative of a function at minimum (inside the
domain) is zero, and the sensitivity to parameters is given by higher order
derivatives. In the case of back-and-forth motions, the efficiency drops
relatively quickly as the angle of attack deviates from the optimal angle,
whereas in the case of up-and-down motions, the minimum is much more flat.
Fig. 4C shows the iso-surfaces
of 1/*P*^{*} as a function of parameters of the upstroke,
(ϵ, α_{u}, β_{u}) for a givenα
_{d}. Their shapes near the maximum have a tube-like structure.
The longitudinal direction of the tube corresponds to the multiple solutions
found here.

Among the parameters in the quasi-steady force model, the one that is most
sensitive to *Re* is *C*_{D}(0);
*C*_{D}(0)∼1/. To
investigate the effect of *Re, C*_{D}(0) is varied from 0.3 to
0.003, which corresponds to *Re* from ∼10^{2} to∼
10^{6}, estimated using the Blasius theory of flow past a plate
(Glauert, 1947). To see the
effect of wing shape, the calculation was repeated using ten randomly chosen
published lift–drag characteristics for airfoils in NACA-4digit, DH and
Xfoil-series (Selig, 2002).
For low Reynolds number plates, *C*_{L}(π/4) and
*C*_{D}(π/2) are varied between 1 and 2.5, to simulate the
effect of the sharpness of the wing tip
(Wang, 2000). The extended
tube structure is found for all tested Reynolds numbers and wing shapes.

## Concluding remarks

The above analysis was partly motivated by the question concerning the relative efficiency of flapping and steady wing motions that support the same weight. The two-stroke model suggests that at the limit where the lift and drag are described by the translational quasi-steady forces, the most efficient motion to support a given weight is the steady wing motion at the optimal angle of attack. There are multiple flapping motions that are very close to the optimum. The efficient two-stroke motions have in common that the downstroke is a gliding motion at an arbitrary angle of attack and the upstroke operates near the optimal angle of attack. We are currently investigating if flapping flight can be more efficient than the steady wing motion when the unsteady aerodynamics effects are included.

Another motivation came from our interest in understanding whether insects are aerodynamically efficient; specifically, whether hovering insects have found some energy-minimizing wing kinematics, given that hovering is an energy-demanding mode of flight. The idea of optimization in biological systems is open to debate. Without testable predictions, however, it is difficult to make progress. Using the same criterion as discussed here, various published wing strokes were examined (Berman and Wang, 2007), for fruit flies (Ennos, 1989; Fry et al., 2003), a bumblebee (Dudley and Ellington, 1990) and a hawkmoth (Willmott and Ellington, 1997), and it was found that some of the specific features of the predicted energy-minimizing hovering kinematics, e.g. the frequency and the wing stroke pattern, are qualitatively and quantitatively similar to the previously observed data. This, however, does not imply that all hovering insects fly using a single pattern of wing motion. Optimal wing motion depends on the wing morphology and lift–drag characteristics. Even for a specific wing, there can be many solutions that are very close to optimal, as indicated here with a four-parameter model. This multiplicity of solutions does not contradict the idea that energy-minimizing shapes the wing kinematics. Instead, we should expect variations in the observed wing kinematics that have comparable efficiency near the optimal.

Finally, to comment on the transition between the up- and downstrokes, the cost of which is assumed to be small. Although muscles in insects are known to be capable of actively pitching the wing (Ellington, 1984; Dickinson et al., 1993), the net power required to pitch the wing in the observed motions of a dragonfly, a fruit fly and a hawkmoth was found to be negative (Bergou et al., 2007). Thus in theory, wing pitching can be aided by aerodynamic torque and does not require additional power. This, and the fact that almost all insects maintain the same leading edge (Fig. 5), suggest that insects may benefit from passive wing pitch reversal during steady flight to simplify control.

## ACKNOWLEDGEMENTS

The work is supported by NSF, AFOSR, and Packard Foundation.

- © The Company of Biologists Limited 2008