#
Comparison of the Average Lift Coefficient ͞C_{L} and Normalized Lift ͞η_{L} for Evaluating Hovering and Forward Flapping Flight

## Abstract

**:**

_{L}for fixed wings. In contrast to its simple and direct application to fixed wings, the equation for ${\overline{C}}_{L}$ requires prior knowledge of the flow field along the wing span, which results in two integrations: along the wing span and over time. This paper proposes an alternate average normalized lift ${\overline{\eta}}_{L}$ that is easy to apply to hovering and forward flapping flight, does not require prior knowledge of the flow field, does not resort to calculus for its solution, and its lineage is close to the basal equation for steady state C

_{L}. Furthermore, the average normalized lift ${\overline{\eta}}_{L}$ converges to the legacy C

_{L}as the flapping frequency is reduced to zero (gliding flight). Its ease of use is illustrated by applying the average normalized lift ${\overline{\eta}}_{L}$ to the hovering and translating flapping flight of bumblebees. This application of the normalized lift is compared to the same application using two widely-accepted legacy average lift coefficients: the first ${\overline{C}}_{L}$ as defined by Dudley and Ellington, and the second lift coefficient by Weis-Fogh. Furthermore, it is shown that the average normalized lift ${\overline{\eta}}_{L}$ has a physical meaning: that of the ratio of work exerted by the flapping wings onto the surrounding flow field and the kinetic energy available at the aerodynamic surfaces during the generation of lift. The working equation for the average normalized lift ${\overline{\eta}}_{L}$ is derived and is presented as a function of Strouhal number, St.

## 1. Introduction

_{L}is a dimensionless number that evaluates the capability of generating lift by a translating fixed wing subjected to steady state aerodynamics [1] (p. 24):

_{L}value, which allows for the comparison of dissimilar flyers (i.e., engineering, as well as biological flyers, with fixed wings during gliding) is obtained by normalizing (dividing) lift L by two variables that are widely accepted in the aerodynamic community: the dynamic pressure ${q}_{\infty}$ (i.e., $\frac{1}{2}\xb7\rho \xb7{{v}_{\infty}}^{2}$) and a reference area, ${s}_{p}$, the wing planform area (note the use of lower case symbol ${s}_{p}$, to be explained later). This ubiquitous and practical Equation (1) has three peculiarities, the

**first peculiarity**is that it uses the kinetic energy per unit volume of the mass of air, ${q}_{\infty}$, as it flows over a static airplane, to normalize lift L.

_{L}.

**second peculiarity**found in the basal Equation (1): the fact that it only accounts for the translation velocity ${v}_{\infty}$ but does not account for the average angular velocity $\overline{\omega}$ of flapping wings. The absence of the average angular velocity impedes the direct application of Equation (1) to evaluate the lift-generating capability of flapping wings. In order to apply the equation to flapping wings, the average angular velocity $\overline{\omega}$ must be inserted in Equation (1) by means of an artificial computational construct, the blade element method (BEM) [4] (p. 347), which result in Equations (2) and (3). This construct consists in dividing the flapping wing into a large number of chordwise elements along its span R. Each of these elements of infinitesimally small width are immersed in a unique local flow field that vary along the length of the flapping wing and must be defined a priori by adding the translation velocity vector ${v}_{\infty}$ (accounted for in Equation (1)) to the local average tangential velocity ${v}_{tg}$ due to average angular velocity $\overline{\omega}$ of the flapping wing (both velocities ${v}_{tg}$ and $\overline{\omega}$ are not accounted for in Equation (1)). The resultant of both of these velocities vary along the spanwise length of the wing r as well as with time t (hence, the integrands in Equations (2) and (3) are integrated with respect to dr and dt). In this way, the BEM is the tool that introduces the effect of the average angular velocity $\overline{\omega}$ of flapping wings in Equation (1).

## 2. The Average Normalized Lift ${\overline{\eta}}_{L}$

_{p}(note upper case S):

_{p}(written in upper case) that differs from the lower case s

_{p}in Equation (1). The original reference area S

_{p}in Equation (1) is found to be its

**third peculiarity**. More on this later.

_{L}in Equation (1). In case of the average lift ${\overline{\eta}}_{L}$ for gliding or soaring flight, $\overline{\omega}$ = 0, we find that the normalized lift ${\overline{\eta}}_{L}$ equals the steady state lift coefficient C

_{L}:

_{L}with one caveat: the definition of the reference area S

_{p}(symbol in upper case) chosen for both ${\overline{\eta}}_{L}$ and C

_{L}is the sum of all planform areas of the aerodynamic surfaces that (i) contribute to the net average lift $\overline{L}$ and (ii) are found (close to) perpendicular to the vector lift $\overline{L}$ (the contribution to lift by the fuselage or body is neglected in this paper). This definition of reference area is considered to be physically proper (upper case S

_{p}) whereas the definition of a reference area that does not consider all surfaces contributing to lift (in a positive or negative sense) is considered a physically improper reference area (lower case s

_{p}), as is the case of the legacy reference area s

_{p}of an airplane in Equation (1) (which does not account for tail or canard surfaces, both contributing to lift). In the case of a bumblebee, the physically proper reference area S

_{p}is its total wing area.

_{k}; (ii) uses only physically proper parameters for normalizing (dividing) the average lift $\overline{L}$ (or average drag $\overline{D}$ and thrust $\overline{T}$, forces not covered in this paper), that is, parameters that have a dominant effect in the generation of lift $\overline{L}$; (iii) is associated with a maximum value ${\overline{\eta}}_{L\text{max}}$ which is usually empirical in nature, and (arguably) close to 1; and (iv) can be read on a stand-alone basis as a “high” or a “low” value.

_{p}and the kinetic energy e

_{k}available at this reference surface S

_{p}during the generation of the average net lift $\overline{L}$ [5]. This ratio w/e

_{k}is made apparent by rewriting Equation (8) as:

_{k}) allows for a novel physical interpretation of the normalized lift, an interpretation that is shared with all physically proper lift coefficients C

_{L}, as applied, say, to a fixed-winged airplane (ω = 0 in Equation (11)) as it accelerates gradually during straight and level flight as it generates a constant amount of lift L (equal to its weight W) while gradually reducing its angle of attack. In this scenario, the work exerted by the fixed wing, L/ρ·S

_{p}, remains constant (L = W = constant) whereas the kinetic energy $\frac{1}{2}\xb7{{v}_{\infty}}^{2}$ available at the wing gradually increases. Its normalized lift ${\eta}_{L}$ (or lift coefficient C

_{L}) measures the amount of work w done “per kinetic energy available” e

_{k}that is found to reduce gradually as evidenced by the gradual reduction of the angle of attack. In other words, L/ρ·S

_{p}remains constant in the numerator of Equation (11), whereas its denominator gradually increases, resulting in a decrease of the normalized lift ${\eta}_{L}$ (or lift coefficient C

_{L}). This physical concept is applicable to the lift coefficient C

_{L}as long as it is calculated by normalizing lift L by physically proper parameters only. The use of one or more physically improper parameters for calculating C

_{L}will render it also physically improper and unfit for use for comparing different lifting surfaces (say, between flapping wings and rotating cylinders in Magnus effect). At this point, and possibly addressing the possible question raised by the reader on the purpose or validity of comparing such differing lifting systems, it is argued that the usefulness of a figure of merit may be seen to increase if these comparisons, however unlikely, are allowed as meaningful (in the same way the efficiency η of, say, the Otto cycle and a jet engine’s Brayton cycle can be compared in thermodynamics). The use of physically improper parameter(s) will result in physically improper legacy coefficients C

_{L}and C

_{Do}that do not allow for such meaningful comparisons, as is the case when comparing the lift coefficient C

_{L}of different aircraft configurations (e.g.; flying wing against tail-configured aircraft) or when comparing the parasite drag coefficient C

_{Do}of airplanes of different wing areas (e.g.; F-104 Starfighter against B-58 Hustler). When using these legacy coefficients, meaningful comparisons can still be made by limiting the comparison of C

_{L}to airplanes of same configuration (flying wing against flying wing), or comparing the C

_{Do}of airplanes with same physically improper reference area s

_{p}[5].

**third peculiarity**of Equation (1): as mentioned above, a valid side-by-side comparison of the normalized lift ${\eta}_{L}$ of steady state lift systems (i.e., fixed-wing aircraft) as well as the average normalized lift ${\overline{\eta}}_{L}$ for time-dependent lift systems (i.e., bumblebees) requires a consistent, physically proper reference area: the reference surface S

_{p}(upper case S) in Equation (5) (and onwards) is the total planform area found (close to) perpendicular and contributing to the net average lift $\overline{L}$. As discussed above, an expected application of a dimensionless coefficient, be it the lift coefficient C

_{L}, the normalized lift ${\eta}_{L}$ or its average value ${\overline{\eta}}_{L}$ is the comparison of the ability of generating lift L by various types of lift systems, be these designed by engineers (i.e., tail or canard-configured airplanes, lift rotors, ornithopters) or researched by biomechanicists (i.e., flapping wings of bumblebees). As mentioned, the possibility of a side-by-side comparison of these differing systems has a valuable cross-pollination potential that unfortunately is not currently possible as the definition of a reference area selected for normalizing steady-state lift L of aircraft (with a reference area represented by a lower case s

_{p}in Equation (1)) is not consistent with the definition of a reference area used for normalizing the time-dependent lift $\overline{L}$ in biological flight (with a reference area represented by an upper case S

_{p}in Equation (5) and onwards). The average lift coefficient ${\overline{C}}_{L}$ of a bumblebee is obtained by normalizing its lift $\overline{L}$ by all the aerodynamic surfaces contributing to its generation, an all-inclusive definition made explicit by the use of the upper case symbol, S

_{p}, as shown in Equation (5). In contrast, and here is the third peculiarity of C

_{L}in Equation (1), the reference area used for a tail or canard-configured airplane considers only the main wing planform s

_{p}. This definition of the legacy s

_{p}, suggested by Munk in 1923 [6], excludes the tail surface and so, neglects its contribution to the net lift L (usually a negative one due to stability purposes) as well as the canard surface (and so, neglects its contribution to the net lift L, always a positive one). This non-inclusive definitions of reference area is a third peculiarity of Equation (1) that results in an physically improper parameter, and is made explicit in this paper by choosing for a lower case symbol, s

_{p}, as shown in Equation (1).

_{p}(as tail and canard areas are not accounted for). This results in an “inflated” wing loading (as s

_{p}< S

_{p}, so L/s

_{p}> $\overline{L}/{S}_{p}$) for the tail and canard-configured airplanes when compared to a bumblebee. This larger wing loading, when divided by the density ρ (as per Equation (11)) results, again, in an “inflated” work w exerted by the tail and canard-configured airplanes, which in turn results in an inflated ${\overline{\eta}}_{L}$ (and ${\overline{\eta}}_{L\text{max}}$) when compared to a bumblebee. This inflated value can be mistakenly reported as a result of a Reynolds number effect but is, instead, due to an inconsistency in the definition of reference areas. That an increase in the Reynolds number has an effect of an increase in C

_{L}

_{max}is not in question: what is highlighted here is a significant contribution towards an increase in the lift coefficient C

_{L}(and C

_{L}

_{max}) that is a result of a more mundane problem: the neglect of the tail and canard areas. If comparisons between biological flyers and aircraft are necessary, the reader is encouraged to compare their legacy C

_{L}(and C

_{L}

_{max}) values using flying wings instead of tail and canard-configured airplanes. In other words: the comparison of the capability of generating lift by tail and canard-configured airplanes on one side and bumblebees on the other may be flawed due to the use of inconsistent definition of their reference areas that, by neglecting a large percentage of their lifting areas that contribute to net lift (≈ tail and canard are typically 20% of the total lifting planform) invalidates a meaningful comparison between lift coefficients, as results show an overestimate of the lift capability of tail and canard-configured by, typically, 20%. Although not related to flapping flight, the above-described situation also arises when comparing the (inevitably lower) C

_{L}

_{max}of a flying wing with the C

_{L}

_{max}of a tail or canard-configured aircraft. The normalized lift ${\eta}_{L}$ is a figure of merit that is not configuration-dependent and allows for the meaningful comparison of a large variety of lifting systems due to its use of a consistent, physically proper definition of reference area S

_{p}[5].

_{k}, that is, energies per unit mass.

^{2}, a value found in [7] (p. 251, Figure 9f). The 1/3 value is what Weis-Fogh calls the shape factor for the second moment of the area, σ [3] (p. 173, Table 1, first row). The second case is when the center of gravity of the wing can be calculated and is not found to be at (or close to) R/2 on the wing but at a distance, say, dCG, from the axis of rotation. In this case, the rod substituting the wing will be of length 2·dCG, and its specific moment of inertia I/m of the wing becomes ⅓·(2·dCG)

^{2}.

^{2}, and $\overline{\omega}$ is replaced by 2·f·Φ, and the fraction ½ is made a common factor and placed outside of the parentheses. With these changes made in the denominator of Equation (11), we define the total wing velocity V

_{w}of a flapping wing as:

_{tt}of the wing tip (subscript tt stands for tip, tangential) during a stroke. Replacing in Equation (12), the total velocity V

_{w}is:

_{∞}is made a common factor and, when taken out of the parentheses, the above expression is written as a function of the velocity ratio ${v}_{tt}/{v}_{\infty}$ that equals the Strouhal number, St [8]:

_{w}is based on kinetic energy considerations and varies from Lentink and Dickinson’s definition of the characteristic speed U, which derives from the kinematics of the flapping wing [9] (p. 2695).

_{w}and the Strouhal number St:

_{0}, the sum of static and dynamic pressure. Note that for the translating flight of fixed wings (i.e., gliding flight), the flapping frequency f is 0, and so, the Strouhal number St is zero, and the total velocity V

_{w}is then reduced to the freestream velocity at infinity, V

_{w}= ${v}_{\infty}$, in Equation (14). Furthermore, the total dynamic pressure Q is reduced to the dynamic pressure ${q}_{\infty}$ in Equation (15).

_{L}and the time-dependent average normalized lift ${\overline{\eta}}_{L}$ is evaluated by the ratio ${C}_{L}/{\overline{\eta}}_{L}$, or the ratio Equation (1)/Equation (16):

_{L}(f→0, then St→0 and ${C}_{L}/{\overline{\eta}}_{L}$→1). Equation (17) can be used to advantage to calculate the average normalized lift ${\overline{\eta}}_{L}$ in two steps: the first step calculates the coefficient C

_{L}for the steady state flight (by assuming extended wings and simply not considering its flapping kinematics) using Equation (1). The second step “corrects” C

_{L}for the time-dependent effects of flapping by dividing the steady state C

_{L}by 1 + ⅓·St

^{2}. The lift coefficient C

_{L}in Equation (17) during flapping flight can be interpreted as the hypothetical steady-state lift coefficient C

_{L}required from the extended, non-flapping wings as they generate an (unrealistic) lift L equal to the weight of the flyer as it translates at the same forward speed ${v}_{\infty}$ as the actual flapping flyer. This steady state C

_{L}is unrealistic as the wings will stall at a much lower value. Correcting this steady-state fictitious C

_{L}value by dividing it (1 + ⅓·St

^{2}) results in the average normalized lift ${\overline{\eta}}_{L}$ of the flapping wings of the flyer. A quasi-steady analysis of flapping flight can be contemplated when the values of the steady state lift coefficient C

_{L}and the corresponding average normalized lift ${\overline{\eta}}_{L}$ are close (i.e.; C

_{L}≈$\text{}{\overline{\eta}}_{L}$). More on this subject in Section 5.

_{w}defined in Equation (14) can be used to advantage to characterize the Reynolds number of flapping wings of characteristic chord c, surrounded by the air of kinematic viscosity, υ:

_{ss}(the subscript ss stands for steady state) contained in the leftmost parentheses, and the second step corrects Re

_{ss}for flapping effects by multiplying it by (1 + ⅓·St

^{2})

^{½}. A closely-related approach to evaluating the Reynolds number of flapping wings has been suggested by Lentink and Dickinson [9] (p. 2696).

_{w}or the total dynamic pressure, Q,

_{w}, Q, ${\overline{\eta}}_{L}$ and Re can be calculated by “correcting” the corresponding steady-state parameters ${v}_{\infty}$, ${q}_{\infty}$, C

_{L}and Re

_{ss}by the term (1 + ⅓·St

^{2}):

^{2}), the ratio of total dynamic pressure and dynamic pressure, Q/q

_{∞}, and the ratio of the Reynolds number of a flapping wing and the corresponding steady state Reynolds number of the same wing, Re/Re

_{ss}, vary with Strouhal number, St:

_{D}

_{max}, and so, cannot be read as a “high” or low” value.

## 3. Evaluation of ${\overline{C}}_{L}$ and ${\overline{\eta}}_{L}$ of Hovering Bumblebees

_{p}and wing root-to-tip length R for the three bumblebees BB01, BB02, and BB03 are presented in Table 3 and were obtained from Dudley and Ellington [10] (p. 32, Table 1):

^{3}at sea level, and the kinematic viscosity υ is assumed to be 1.46 × 10

^{−5}m

^{2}/s, corresponding to the according to the standard atmosphere [11].

## 4. Evaluating ${\overline{C}}_{L}$ and ${\overline{\eta}}_{L}$ of Forward Flying Bumblebees

_{ss}corresponding to the steady state for the same wing as it flies at the same translating velocity ${v}_{\infty}$ as the flapping wing).

## 5. Evaluating the Aerodynamics of Bumblebee BB01

_{L}are calculated using the same physically proper reference area S

_{p}, as defined in Section 2 (i.e.; the total wing area of the bumblebee). The first step calculates the steady state (non-flapping) lift coefficient C

_{L}of the bumblebee using Equation (1), using the following information: its weight W of 0.001715 N, its reference area S

_{p}of 0.000106 m

^{2}, the density at sea level of 1.23 kg/m

^{3}, and a forward velocity ${v}_{\infty}$ of 2.5 m/s. The resulting steady state lift coefficient C

_{L}equals 4.2, an obviously unrealistic value that is much higher than the maximum value C

_{L}

_{max}that can be possibly reached during steady state at these low (or any) Reynolds numbers (the steady state value for C

_{L}

_{max}at this Reynolds numbers is likely < 1). The second step modifies this steady state lift coefficient C

_{L}by dividing it by the “correction factor” (1 + ⅓·St

^{2}) to account for the effects of time-dependent flow. This correction factor is 5.08, where the Strouhal number St is 3.5 $\left(St=\frac{2\xb7f\xb7\mathsf{\Phi}\xb7R}{{v}_{\infty}}=\frac{2\xb7152\xb72.18\xb70.0132}{2.5}\right)$. The resulting average normalized lift ${\overline{\eta}}_{L}$ is 0.83 (= 4.2/5.08).

_{L}of 4.2 and intersecting the isoline corresponding to a constant Strouhal number St of 3.5, and reading the resulting average normalized lift of ${\overline{\eta}}_{L}$ on the vertical axis: 0.83.

_{L}calculated using Equation (1). In other words, for flight conditions where the Strouhal number St is equal or less than 0.2, then ${\overline{\eta}}_{L}$ ≈ C

_{L}and so, it can be estimated by C

_{L}, and the actual average normalized lift ${\overline{\eta}}_{L}$ is smaller than C

_{L}by 1.33%. If St = 0.3, ${\overline{\eta}}_{L}$ is smaller than C

_{L}by 2.91% and for St = 1, ${\overline{\eta}}_{L}$ is smaller than C

_{L}by 33%.

_{ss}for the corresponding steady state, non-flapping flight, Re/Re

_{ss}, is given by Equation (21). Following a similar aforementioned two-step procedure, the actual Reynolds number Re of flapping wings is calculated by first calculating its steady state Reynolds number Re

_{ss}for the wing of BB01 of chord c of 0.002 m (span/aspect ratio = R/AR = 0.0132/6.56), flying at a forward velocity ${v}_{\infty}$ of, say, 2.5 m/s, at sea level. This results in a steady state Reynolds number Re

_{ss}of 344, and when multiplied by (1+ ⅓·St

^{2})

^{½}with St = 3.5, it results in Re of 777:

^{2})

^{½}in Equation (21) is a multiplier that converts the Reynolds number from steady to time-dependent values. The multiplier can be computed graphically by using Figure 3: the Strouhal number St of 3.5 is entered on the abscissa (horizontal axis) and intercepting the curve, one reads the value for Re/Re

_{ss}of 2.26 on the ordinate axis. Multiplying Re

_{ss}by this number results in Re.

_{ss}as Re ≈ Re

_{ss}.

## 6. Conclusions

_{L}is evidenced by making the flapping frequency f approach zero in Equation (8): when f → 0, then $\overline{\omega}$ → 0, and ${\overline{\eta}}_{L}$. Furthermore, a quasi-steady regime for flapping flight can be defined quantitatively as a function of the Strouhal number. This region, shown in Figure 2 is bounded by 0 < St < 0.2, where the maximum difference between ${\overline{\eta}}_{L}$ and C

_{L}in this region is never to exceed 1.33%. This suggestion of the quasi-steady regime boundary may only be accepted if this difference of 1.33% is acceptable. A reason the particular upper boundary for quasi-steady flight was suggested is that it coincides with the lower boundary of the Strouhal number that defines the region for high power efficiency for flying and swimming animals during cruise, namely, 0.2 < St < 0.4, as documented by Taylor et al. [8]. The equations for the average normalized lift ${\overline{\eta}}_{L}$ presented in this paper can also be applied to underwater locomotion for the calculation of the average thrust ${\overline{\eta}}_{T}$ and drag ${\overline{\eta}}_{D}$ by using the appropriate physically proper parameters (i.e.; the reference planform area of the caudal fin for calculating ${\overline{\eta}}_{T}$, and the reference frontal area for calculating ${\overline{\eta}}_{D}$) [5].

_{k}available at the wing, a meaning that makes the normalized lift ${\overline{\eta}}_{L}$ independent of the configuration or type of lifting system. The average normalized lift ${\overline{\eta}}_{L}$ of a bumblebee can be compared meaningfully to the normalized lift ${\eta}_{L}$ of, say, a tail or canard-configured airplane, or the normalized lift ${\eta}_{L}$ of a rotating cylinder in Magnus effect, a lift rotor, a quadcopter or an ornithopter.

_{p}as the sum of all aerodynamic surfaces contributing to its net lift L, that is, including horizontal tail and/or canard planform areas in the case of aircraft). Here, the term “maximum operating ${\overline{\eta}}_{L}$” is a transparent means by the author of staying away from ${\overline{\eta}}_{L\text{max}}$ (stall). Additionally, it may be observed that the hovering bumblebees may not be flying at their ${\overline{\eta}}_{L\text{max}}$ as they may have some energy reserve for an upward vertical acceleration. In the same way, airplanes may have to fly at a higher normalized lift to effectively experience stall.

_{N}can be found in [12].

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Glossary

A | amplitude (distance travelled) by the wingtips over a wing stroke |

c | mean chord of wing |

C_{L} | steady-state lift coefficient (i.e., during gliding or soaring) |

${\overline{C}}_{LDE}$ | average lift coefficient derived by Dudley and Ellington |

${\overline{C}}_{LW-F}$ | Weis-Fogh’s quick estimate of average lift applicable to hovering flight only |

dCG | distance between spanwise location of center of gravity of wing and rod to axis of rotation |

e_{k} | specific kinetic energy per unit mass of flyer at the flapping wing |

f | stroke frequency during flapping |

I/m | specific moment of inertia of flapping wing about its end (shoulder or hinge) |

L | lift |

L_{b} | Lift on the body |

$\overline{L}$ | average lift |

m | mass of the flyer or lifting system, mass of surrounding fluid (air) |

f | flapping frequency |

q_{∞} | dynamic pressure, ½·ρ·v_{∞}^{2} of a moving mass of fluid (air) |

Q | total dynamic pressure, not to be confused with the sum of static and dynamic pressure, p_{o} |

R | root-to-tip length of wing. Length of rod replacing a wing, used in calculation of I/m |

r | a variable of integration representing the distance from wing root to a given wing station along the semispan, 0 < r < R |

Re_{ss} | Reynolds number for steady state flight |

Re | Reynolds number of a flapping wing |

s_{p} | physically improper (lower case) reference wing planform area (subscript p stands for planform). Excludes other aerodynamic surfaces contributing to the net lift L or average net lift $\overline{L}$. Upper case S_{p} is physically proper reference area |

St | Strouhal number |

S_{ref} | physically proper reference area used to normalize lift L for obtaining ${\overline{\eta}}_{L}$. It is obtained by adding all planform areas that contribute to the generation of lift L, ΣS_{ref i} |

T | period, the duration of one complete flapping cycle or wingbeat |

t | time, variable of integration |

v_{tg} | tangential velocity due to the angular velocity of the flapping wing at a given chordwise wing element during the application of the blade element analysis |

v_{tt} | tangential velocity of flapping wing at the tip |

v_{∞} | freestream velocity at infinity relative to a static flyer, velocity of translating flight while inmersed in static mass of air |

V_{w} | total velocity |

W | weight of the flyer during equilibrium flight (forward flight or hover), equal to average lift, $\overline{L}$ |

v_{r}(r,t) | relative velocity |

β | stroke plane angle |

ρ | density of air at sea level, 1.23 kg/m^{3} |

η_{L} | normalized lift of fixed wing and propeller blades |

${\overline{\eta}}_{L}$ | average normalized lift of flapping wings |

Φ | stroke angle in radians during downstroke or upstroke |

υ | kinematic viscosity, 1.46 × 10^{−5} m^{2}/s |

σ | shape factor for the second moment of the wing area [3] |

$\overline{\omega}$ | average flapping angular speed |

Ψ | direction of the relative velocity vector |

## References

- Anderson, J.D. Fundamentals of Aerodynamics, 5th ed.; McGraw Hill: Pennsylvania Plaza, NY, USA, 2011. [Google Scholar]
- Dudley, R.; Ellington, C.P. Mechanics of forward flight in bumblebees. II. Quasi-steady lift and power requirements. J. Exp. Biol.
**1990**, 148, 53–88. [Google Scholar] - Weis-Fogh, T. Quick Estimates of flight fitness in hovering animals, including novel mechanisms for lift production. J. Exp. Biol.
**1973**, 59, 169–230. [Google Scholar] - McCormick, B.W. Aerodynamics, Aeronautics and Flight Mechanics; John Wiley & Sons: Hoboken, NJ, USA, 1979. [Google Scholar]
- Burgers, P. Dimensionally and Physically Proper Lift, Drag and thrust-related numbers as figures of merit, η
_{L}, η_{D}and η_{T}. J. Aerosp. Eng.**2016**. in review. [Google Scholar] - Munk, M.M. General Biplane Theory; NACA TR 151; National Advisory Committee for Aeronautics: Washington, DC, USA, 1923; p. 495. [Google Scholar]
- Resnick, R.; Halliday, D.; Krane, K.S. Physics, 4th ed.; Wiley: Hoboken, NJ, USA, 1992; Volume 1. [Google Scholar]
- Taylor, G.K.; Nudds, R.L.; Thomas, A.L.R. Flying and swimming animals cruise at a Strouhal number tuned for high power efficiency. Nature
**2003**, 425, 707–711. [Google Scholar] [CrossRef] [PubMed] - Lentink, D.; Dickinson, M.H. Biofluiddynamic scaling of flapping, spinning and translating fins and wings. J. Exp. Biol.
**2009**, 212, 2691–2704. [Google Scholar] [CrossRef] [PubMed] - Dudley, R.; Ellington, C.P. Mechanics of forward flight in bumblebees. I. Kinematics and morphology. J. Exp. Biol.
**1990**, 148, 19–52. [Google Scholar] - U.S. Government Printing Office. U.S. Standard Atmosphere, 1976; U.S. Government Printing Office: Washington, DC, USA, October 1976. [Google Scholar]
- Burgers, P.; Alexander, D. Normalized Lift: An energy interpretation of the lift coefficient simplifies comparisons of the lifting ability of rotating and flapping surfaces. PLoS ONE
**2012**, 7, e36732. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**The average lift coefficient ${\overline{C}}_{L}$ (dashed lines) and average normalized lift ${\overline{\eta}}_{L}$ (continuous lines) of the three bumblebees during hover (along the ordinate axis, at ${v}_{\infty}$ = 0) and forward flight (for ${v}_{\infty}$ > 0). Grey labels relate to abnormal data encountered by Dudley and Ellington.

**Figure 2.**The average normalized lift ${\overline{\eta}}_{L}$ of flapping wings is related to their steady-state lift coefficient C

_{L}of fixed wings by the isolines corresponding to various Strouhal numbers (0.2 < St < 5 shown). The quasi-steady region, ${\overline{\eta}}_{L}$ ≈ C

_{L}, is bound by St < 0.2.

**Figure 3.**The ratio of the Re of flapping wings and the Re

_{ss}for steady state wings, Re/Re

_{ss}, as a function of the Strouhal number.

Time-Dependent Variable | Time-Dependent Variables as a Function of Steady State Parameters | Equation No. |
---|---|---|

Total velocity, V_{w} | V_{w} = v_{∞}·(1 + ⅓·St^{2})^{½} | 14 |

Total dynamic pressure, Q | Q = q_{∞}·(1 + ⅓·St^{2}) | 15 |

Average normalized Lift, ${\overline{\eta}}_{L}$ | ${\overline{\eta}}_{L}$ = C_{L}·(1 + ⅓·St^{2})^{−1} | 17 |

Reynolds number, Re | Re = Re_{ss}·(1 + ⅓·St^{2})^{½} | 18 |

**Table 2.**Strouhal number effect on total dynamic pressure Q and Reynolds number Re of flapping wings.

St | 1 + ⅓·St^{2} | Q/q_{∞} | Re/Re_{ss} |
---|---|---|---|

0 | 1.00 | 1.00 | 1.00 |

1 | 1.33 | 1.33 | 1.15 |

2 | 2.33 | 2.33 | 1.53 |

3 | 4.00 | 4.00 | 2.00 |

4 | 6.33 | 6.33 | 2.52 |

5 | 9.33 | 9.33 | 3.06 |

6 | 13.00 | 13.00 | 3.61 |

7 | 17.33 | 17.33 | 4.16 |

8 | 22.33 | 22.33 | 4.73 |

9 | 28.00 | 28.00 | 5.29 |

10 | 34.33 | 34.33 | 5.86 |

ID | W (N) | S_{p} (m^{2}) | R (m) |
---|---|---|---|

BB01 | 0.00172 | 0.00011 | 0.0132 |

BB02 | 0.0018 | 0.0001 | 0.0137 |

BB03 | 0.00583 | 0.0137 | 0.0154 |

ID | f (Hz) | Φ (rad) | $\overline{\omega}$ (1/s) |
---|---|---|---|

BB01 | 155 | 2.02 | 627.57 |

BB02 | 147 | 1.82 | 533.61 |

BB03 | 166 | 2.27 | 753.23 |

**Table 5.**Averages of the two legacy average lift coefficients ${\overline{C}}_{L}$ and the average normalized lift ${\overline{\eta}}_{L}$ for the three bumblebees during hover. Note lower average value of 1.29 for average normalized lift ${\overline{\eta}}_{L}$.

ID | ${\overline{C}}_{LDE}$ | ${\overline{C}}_{LW-F}$ | ${\overline{\eta}}_{L}$ |
---|---|---|---|

BB01 | 1.2 | 1.87 | 1.15 |

BB02 | 2.1 | 2.35 | 1.45 |

BB03 | 2.65 | 2.09 | 1.29 |

Average | 1.98 | 2.01 | 1.29 |

**Table 6.**Kinematic data of flapping wings, flight data, Dudley and Ellington’s average lift coefficient ${\overline{C}}_{LDE}$, average normalized lift ${\overline{\eta}}_{L}$ , and the ratio of flapping Re to steady-state Re

_{ss}.

ID | ${v}_{\infty}$ (m/s) | f (Hz) | Φ (rad) | $\overline{\omega}$ (1/s) | v_{tt} (m/s) | St | ${\overline{\eta}}_{L}$ | ${\overline{C}}_{LDE}$ | (1 + ⅓·St^{2})^{½} | Re/Re_{ss} |
---|---|---|---|---|---|---|---|---|---|---|

BB01 | 0 | 155 | 2.02 | 627.57 | – | – | 1.15 | 1.2 | – | – |

1 | 145 | 1.95 | 566.84 | 7.48 | 7.48 | 1.34 | 1.72 | 19.66 | 4.43 | |

2.5 | 152 | 2.18 | 663.18 | 8.75 | 3.50 | 0.83 | 1.28 | 5.09 | 2.26 | |

4.5 | 144 | 1.80 | 517.70 | 6.83 | 1.52 | 0.74 | 1.15 | 1.77 | 1.33 | |

BB02 | 0 | 147 | 1.82 | 533.61 | – | – | 1.45 | 2.1 | – | – |

1 | 132 | 1.73 | 456.13 | 6.25 | 6.25 | 1.84 | 2.18 | 14.02 | 3.74 | |

2.5 | 132 | 2.01 | 529.84 | 7.26 | 2.90 | 1.09 | 1.8 | 3.81 | 1.95 | |

4.5 | 143 | 1.66 | 474.17 | 6.50 | 1.44 | 0.75 | 1.05 | 1.69 | 1.30 | |

BB03 | 0 | 166 | 2.27 | 753.23 | – | – | 1.29 | 2.65 | – | – |

1 | 157 | 2.22 | 695.95 | 10.72 | 10.72 | 1.47 | 3.25 | 39.29 | 6.27 | |

2.5 | 141 | 1.68 | 472.46 | 7.28 | 2.91 | 2.42 | 2.1 | 3.82 | 1.96 |

**Table 7.**The Dudley and Ellington’s average lift coefficient ${\overline{C}}_{LDE}$, and the average normalized lift ${\overline{\eta}}_{L}$, and corresponding averages for forward flight.

ID | ${\overline{C}}_{LDE}$ | ${\overline{\eta}}_{L}$ | ${\overline{C}}_{LDE}$ | ${\overline{\eta}}_{L}$ | ${\overline{C}}_{LDE}$ | ${\overline{\eta}}_{L}$ |
---|---|---|---|---|---|---|

1 m/s | 2.5 m/s | 4.5 m/s | ||||

BB01 | 1.72 | 1.34 | 1.28 | 0.83 | 1.15 | 0.74 |

BB02 | 2.18 | 1.84 | 1.8 | 1.09 | 1.05 | 0.75 |

BB03 | 3.25 | 1.47 | 2.1 | – | – | – |

Average | 2.38 | 1.55 | 1.73 | 1.89 | 1.89 | 1.89 |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Burgers, P.
Comparison of the Average Lift Coefficient ͞*C _{L}* and Normalized Lift ͞

*η*for Evaluating Hovering and Forward Flapping Flight.

_{L}*Aerospace*

**2016**,

*3*, 24. https://doi.org/10.3390/aerospace3030024

**AMA Style**

Burgers P.
Comparison of the Average Lift Coefficient ͞*C _{L}* and Normalized Lift ͞

*η*for Evaluating Hovering and Forward Flapping Flight.

_{L}*Aerospace*. 2016; 3(3):24. https://doi.org/10.3390/aerospace3030024

**Chicago/Turabian Style**

Burgers, Phillip.
2016. "Comparison of the Average Lift Coefficient ͞*C _{L}* and Normalized Lift ͞

*η*for Evaluating Hovering and Forward Flapping Flight"

_{L}*Aerospace*3, no. 3: 24. https://doi.org/10.3390/aerospace3030024