Minimizing Induced Drag with Lift Distribution and Wingspan

Phillips, Warren F.; Hunsaker, Douglas F.; Joo, James J.

doi:10.2514/1.c035027

Cited by 40 publications

(80 citation statements)

References 27 publications

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“…Spanwise chord profiles matching the wing and tail spans of the measured glides were calculated from point clouds, excluding the head, from earlier glides using high-speed video photogrammetric methods, and were fitted with 50 Fourier terms to provide a closethough constrained to be symmetrical about the centre linerepresentation of the chord profile. This technique allows classical aerodynamic methods (Munk, 1923;Prandtl, 1921;Houghton et al, 2016, Phillips et al, 2019 to be applied to determine the associated downwash profiles given the assumption that profile drag is minimized if all sections operate at constant lift coefficient (and the lift coefficient is sufficient to support body weight). Shape parameters and their derivatives were obtained from planforms measured during an earlier study, best matched to landmarks at wing and tail tips measured in the current study.…”

Section: Relating Drag Minimization Predictions To Downwash Profilesmentioning

confidence: 99%

“…While an elliptical loading distribution provides the theoretical minimum induced drag for a constrained wing span, other loading distributions are optimal given different constraints. Various structural, geometrical and weight considerations, along with passive yaw stability, may be important in aircraft design, leading to a range of non-elliptical loading distributions providing theoretical optima for minimizing induced drag (Prandtl, 1933;Phillips et al, 2019). The optimal loading distributions with such constraints tend to be more 'bell shaped', with a bias in loading towards central sections of the vehicle.…”

Section: Further Caveats and Comments A Note On Passive Longitudinal mentioning

confidence: 99%

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High aerodynamic lift from the tail reduces drag in gliding raptors

Usherwood

Cheney

Song

et al. 2020

Journal of Experimental Biology

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Many functions have been postulated for the aerodynamic role of the avian tail during steady-state flight. By analogy with conventional aircraft, the tail might provide passive pitch stability if it produced very low or negative lift. Alternatively, aeronautical principles might suggest strategies that allow the tail to reduce inviscid, induced drag: if the wings and tail act in different horizontal planes, they might benefit from biplane-like aerodynamics; if they act in the same plane, lift from the tail might compensate for lift lost over the fuselage (body), reducing induced drag with a more even downwash profile. However, textbook aeronautical principles should be applied with caution because birds have highly capable sensing and active control, presumably reducing the demand for passive aerodynamic stability, and, because of their small size and low flight speeds, operate at Reynolds numbers two orders of magnitude below those of light aircraft. Here, by tracking up to 20,000, 0.3 mm neutrally buoyant soap bubbles behind a gliding barn owl, tawny owl and goshawk, we found that downwash velocity due to the body/tail consistently exceeds that due to the wings. The downwash measured behind the centreline is quantitatively consistent with an alternative hypothesis: that of constant lift production per planform area, a requirement for minimizing viscous, profile drag. Gliding raptors use lift distributions that compromise both inviscid induced drag minimization and static pitch stability, instead adopting a strategy that reduces the viscous drag, which is of proportionately greater importance to lower Reynolds number fliers.

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Section: Relating Drag Minimization Predictions To Downwash Profilesmentioning

confidence: 99%

Section: Further Caveats and Comments A Note On Passive Longitudinal mentioning

confidence: 99%

High aerodynamic lift from the tail reduces drag in gliding raptors

Usherwood

Cheney

Song

et al. 2020

Journal of Experimental Biology

View full text Add to dashboard Cite

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“…More recently, Phillips et al [12,38] revisited Prandtl's 1933 analysis [14] and relaxed many of his main assumptions, including the assumption that the proportionality coefficient is spanwise invariant and independent of the wing geometry. Instead, Phillips et al [12,38] related the proportionality coefficient to the local wing dimensions, wing-structure shape, and the wing-structure material. Thus, the development given by Phillips et al [12] includes the effects of the wing-structure and the chord distribution.…”

Section:   Bmentioning

confidence: 99%

“…From classical lifting-line theory, the spanwise lift distribution can be written in terms of a Fourier series. Although this series is generally written in an alternate form, here we shall use the dimensionless form [12]…”

Section: Introductionmentioning

confidence: 99%

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Minimum Induced Drag for Tapered Wings Including Structural Constraints

Taylor

Hunsaker

2020

Journal of Aircraft

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Minimum‐series twist distributions for yawing‐moment control during pure roll*

2021

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The main wings of most aircraft produce adverse yaw during roll. In order to control the lateral direction of the aircraft during the roll, the rudder is often mixed with the aileron. Lifting-line theory is used here to develop spanwise lift distributions that require the minimum number of terms in the Fourier-series solution for controlling the yawing moment during pure rolling motion using only the main wing. It is shown that the yawing moment can be controlled for arbitrary rolling moments and/or rolling rates by adding symmetric twist in the main wing. The induced-drag penalty for using this method to control the yawing moment is significant and discussed in detail. For example, it is shown that if zero yawing moment is prescribed, the induced drag can increase by 108% for a prescribed rolling moment or by 300% during a steady rolling rate relative to the induced drag in steady level flight. Because this is the minimum-series solution, it does not represent the solution for yaw control with minimum induced drag, since more terms could be used in the Fourier series describing the lift distribution to control yaw with less induced drag. However, the solutions presented here can be useful for aircraft with continuous trailing-edge technologies that are Nomenclature:, Fourier coefficients in the lifting-line solution for the section-lift distribution, Eq. (1); , coefficients in the decomposed Fourier-series solution to the lifting-line equation related to planform, Eqs. (8) and ( 9); b, wingspan; , coefficients in the decomposed Fourier-series solution to the lifting-line equation related to symmetric baseline twist, Eqs. ( 8) and (10);, induced-drag coefficient; , induced-drag coefficient produced by the elliptic lift distribution for a given aspect ratio and lift coefficient, Eq. (21); , total lift coefficient;̃, section-lift coefficient;̃, , section-lift slope; , rolling-moment coefficient; , yawing-moment coefficient; , , change in yawing moment with rolling moment; ,̄, change in yawing moment with rolling rate; c, section chord length;̄, wing mean chord length; , coefficients in the decomposed Fourier-series solution to the lifting-line equation due to antisymmetric twist related to roll control, Eqs. ( 8) and (11); c root , chord length at wing root; , coefficients in the decomposed Fourier-series solution to the lifting-line equation related to rolling rate, Eqs. (8) and ( 12); , coefficients in the decomposed Fourier-series solution to the lifting-line equation due to symmetric twist related to yaw control, Eqs. (8) and (13); j, term in the infinite series;̃, section lift; N, number of terms retained when truncating the infinite Fourier series; , angular rigid-body rolling rate;̄, dimensionless angular rigid-body rolling rate [pb/(2V ∞ )]; , wing aspect ratio ( 2 ∕ ); , wing taper ratio; , wing planform area; ∞ , freestream airspeed; z, spanwise coordinate relative to the midspan, positive out left wing; , spanwise variation in local geometric angle of attack relative to the freestream; 0 , spanwise varia...

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Minimizing Induced Drag with Lift Distribution and Wingspan

Cited by 40 publications

References 27 publications

High aerodynamic lift from the tail reduces drag in gliding raptors

High aerodynamic lift from the tail reduces drag in gliding raptors

Minimum Induced Drag for Tapered Wings Including Structural Constraints

Minimum‐series twist distributions for yawing‐moment control during pure roll*

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