In 1933, Ludwig Prandtl published a paper in the journal Zeitschrift für Flugtechnik und Motorluftschiffarht concerning minimum induced drag of straight wings. His publication showed that the elliptic lift distribution is not the optimum lift distribution to minimize induced drag in all cases. In fact, the elliptic lift distribution is only optimum for the special case of prescribed gross weight and wingspan. Because wing weight is a function of wingspan and lift distribution, there exists an optimum wingspan and lift distribution for a given net weight. Prandtl obtained an analytic solution for an alternate lift distribution that minimizes induced drag for prescribed gross lift and prescribed moment of inertia of the lift distribution. Unlike many of Prandtl’s other publications, his 1933 paper was never officially published in the English language. The present work includes an electronic typeset of the original German document, as well as a careful English translation. A commentary on the translation and typesetting is also included in an attempt to remove any confusion about the original document.

ABSTRACTBecause the wing-structure weight required to support the critical wing section bending moments is a function of wingspan, net weight, weight distribution, and lift distribution, there exists an optimum wingspan and wing-structure weight for any fixed net weight, weight distribution, and lift distribution, which minimises the induced drag in steady level flight. Analytic solutions for the optimum wingspan and wing-structure weight are presented for rectangular wings with four different sets of design constraints. These design constraints are fixed lift distribution and net weight combined with 1) fixed maximum stress and wing loading, 2) fixed maximum deflection and wing loading, 3) fixed maximum stress and stall speed, and 4) fixed maximum deflection and stall speed. For each of these analytic solutions, the optimum wing-structure weight is found to depend only on the net weight, independent of the arbitrary fixed lift distribution. Analytic solutions for optimum weight and lift distributions are also presented for the same four sets of design constraints. Depending on the design constraints, the optimum lift distribution can differ significantly from the elliptic lift distribution. Solutions for two example wing designs are presented, which demonstrate how the induced drag varies with lift distribution, wingspan, and wing-structure weight in the design space near the optimum solution. Although the analytic solutions presented here are restricted to rectangular wings, these solutions provide excellent test cases for verifying numerical algorithms used for more general multidisciplinary analysis and optimisation.

No AccessEngineering NotesDesigning Wing Twist or Planform Distributions for Specified Lift DistributionsW. F. Phillips and D. F. HunsakerW. F. PhillipsUtah State University, Logan, Utah 84322-4130 and D. F. HunsakerUtah State University, Logan, Utah 84322-4130Published Online:27 Dec 2018https://doi.org/10.2514/1.C035206SectionsRead Now ToolsAdd to favoritesDownload citationTrack citations ShareShare onFacebookTwitterLinked InRedditEmail About References [1] Prandtl L., "Tragflügel Theorie," Nachricten von der Gesellschaft der Wissenschaften zu Göttingen, Geschäeftliche Mitteilungen, Klasse, Germany, 1918, pp. 451–477. Google Scholar[2] Prandtl L., "Applications of Modern Hydrodynamics to Aeronautics," NACA TR-116, June 1921. Google Scholar[3] Glauert H., "The Monoplane Aerofoil," The Elements of Aerofoil and Airscrew Theory, Cambridge Univ. Press, Cambridge, U.K., 1926, pp. 137–155. Google Scholar[4] Multhopp H., "Die Berechnung der Auftriebs Verteilung von Tragflugeln," Luftfahrtforschung, Vol. 15, No. 14, 1938, pp. 153–169. LFFGAV Google Scholar[5] Lotz I., "Berechnung der Auftriebsverteilung beliebig geformter Flugel," Zeitschrift für Flugtechnik und Motorluftschiffahrt, Vol. 22, No. 7, 1931, pp. 189–195. Google Scholar[6] Karamcheti K., "Elements of Finite Wing Theory," Ideal-Fluid Aerodynamics, Wiley, New York, 1966, pp. 535–567. Google Scholar[7] Anderson J. D., "Incompressible Flow over Finite Wings: Prandtl's Classical Lifting-Line Theory," Fundamentals of Aerodynamics, 3rd ed., McGraw–Hill, New York, 2001, pp. 360–387. Google Scholar[8] Bertin J. J., "Incompressible Flow About Wings of Finite Span," Aerodynamics for Engineers, 4th ed., Prentice-Hall, Upper Saddle River, NJ, 2002, pp. 230–302. Google Scholar[9] Katz J. and Plotkin A., "Finite Wing: The Lifting-Line Model," Low-Speed Aerodynamics, 2nd ed., Cambridge Univ. Press, Cambridge, U.K., 2001, pp. 167–183. CrossrefGoogle Scholar[10] Kuethe A. M. and Chow C. Y., "The Finite Wing," Foundations of Aerodynamics, 5th ed., Wiley, New York, 1998, pp. 169–219. Google Scholar[11] McCormick B. W., "The Lifting-Line Model," Aerodynamics, Aeronautics, and Flight Mechanics, 2nd ed., Wiley, New York, 1995, pp. 112–119. Google Scholar[12] Phillips W. F., "Incompressible Flow over Finite Wings," Mechanics of Flight, 2nd ed., Wiley, Hoboken, NJ, 2010, pp. 46–94. Google Scholar[13] Phillips W. F. and Alley N. R., "Predicting Maximum Lift Coefficient for Twisted Wings Using Lifting-Line Theory," Journal of Aircraft, Vol. 44, No. 3, 2007, pp. 898–910. doi:https://doi.org/10.2514/1.25640 LinkGoogle Scholar[14] Phillips W. F., Fugal S. R. and Spall R. E., "Minimizing Induced Drag with Wing Twist, Computational-Fluid-Dynamics Validation," Journal of Aircraft, Vol. 43, No. 2, 2006, pp. 437–444. doi:https://doi.org/10.2514/1.15089 LinkGoogle Scholar[15] Alley N. R., Phillips W. F. and Spall R. E., "Predicting Maximum Lift Coefficient for Twisted Wings Using Computational Fluid Dynamics," Journal of Aircraft, Vol. 44, No. 3, 2007, pp. 911–917. doi:https://doi.org/10.2514/1.25643 LinkGoogle Scholar[16] Phillips W. F., Hunsaker D. F. and Joo J. J., "Minimizing Induced Drag with Lift Distribution and Wingspan," Journal of Aircraft, Sept. 2018, pp. 1–11. doi:https://doi.org/10.2514/1.C035027 Google Scholar Previous article FiguresReferencesRelatedDetailsCited byNovel Approach for Wing Design in Conceptual Overall Aircraft DesignTim Effing, Florian Schueltke, Yves Heuschling, Paul Mauerer and Eike Stumpf19 January 2023Lifting-Line Predictions for Lift and Twist Distributions to Minimize Induced Drag in Ground EffectKyler Church and Douglas F. Hunsaker19 January 2023Simulations of a Bell-Shaped Span-Loaded Swept WingPatrick R. Hammer and Daniel J. 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Hunsaker5 January 2020Revisiting the combinatorics of lifting line and 2D vortex lattice theory9 August 2019 | The Aeronautical Journal, Vol. 123, No. 1265Minimum-Frequency Spanwise Twist for Yawing-Moment Control During RollDouglas F. Hunsaker, Bruno Moorthamers and James J. Joo15 June 2019 What's Popular Volume 56, Number 2March 2019 CrossmarkInformationCopyright © 2018 by Warren F. Phillips and Douglas F. Hunsaker. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. All requests for copying and permission to reprint should be submitted to CCC at www.copyright.com; employ the eISSN 1533-3868 to initiate your request. See also AIAA Rights and Permissions www.aiaa.org/randp. TopicsAerodynamic PerformanceAerodynamicsAeronautical EngineeringAeronauticsAerospace SciencesAirspeedComputational Fluid DynamicsFlight DynamicsFlight MechanicsFluid Dynamics KeywordsLifting Line TheoryMorphing WingAerodynamic AngleAspect RatioUnswept WingsComputational Fluid DynamicsComputingFlight PhasesWing RootAirspeedAcknowledgmentThis work was partially funded by the U.S. Office of Naval Research Sea-Based Aviation program (Grant No. N00014-18-1-2502) with Brian Holm-Hansen as the program officer.PDF Received10 August 2018Accepted11 November 2018Published online27 December 2018

Minimum induced drag for fixed gross weight and wingspan is obtained from the elliptic lift distribution. However, minimum induced drag for steady level flight is not obtained by imposing the constraints of fixed gross weight and wingspan. Because required wing-structure weight is a function of wingspan and lift distribution, there exist an optimum lift distribution and wingspan for a given weight and weight distribution that minimizes the induced drag in steady level flight. This optimum lift distribution can vary significantly from the elliptic lift distribution, depending on the design constraints. Analytic solutions for three such optimum lift distributions are presented for rectangular wings with varying sets of design constraints. These design constraints are 1) weight, maximum stress, and chord length fixed; 2) weight, maximum stress, and wing loading fixed; and 3) weight, maximum deflection, and wing loading fixed. It is shown that each of these optimum lift distributions results in lower induced drag than a fixed elliptic lift distribution for the same design constraints.

It is shown that the smooth-wall boundary conditions specified for commonly used dissipation-based turbulence models are mathematically incorrect. It is demonstrated that when these traditional wall boundary conditions are used, the resulting formulations allow either an infinite number of solutions or no solution. Furthermore, these solutions do not enforce energy conservation and they do not properly enforce the no-slip condition at a smooth surface. This is true for all dissipation-based turbulence models, including the k-{\epsilon}, k-{\omega}, and k-{\zeta} models. Physically correct wall boundary conditions must force both k and its gradient to zero at a smooth wall. Enforcing these two boundary conditions on k is sufficient to determine a unique solution to the coupled system of differential transport equations. There is no need to impose any wall boundary condition on {\epsilon}, {\omega}, or {\zeta} at a smooth surface and it is incorrect to do so. The behavior of {\epsilon}, {\omega}, or {\zeta} approaching a smooth surface is that required to satisfy the differential equations and force both k and its gradient to zero at the wall.

It is shown that the time-dependent aerodynamic forces acting on a flapping airfoil in forward flight are functions of both axial and normal reduced frequencies. The axial reduced frequency is based on the chord length, and the normal reduced frequency is based on the plunging amplitude. Furthermore, the time-dependent aerodynamic forces are related to two Fourier coefficients, which are evaluated here from computational results. Correlation equations for these Fourier coefficients are obtained from a large number of grid- and time-step-resolved inviscid computational-fluid-dynamics solutions, conducted over a range of both axial and normal reduced frequencies. The correlation results can be used to predict the thrust, required power, and propulsive efficiency for airfoils in forward flight with sinusoidal pitching and plunging motion. Within the range of parameters typically encountered in the efficient forward flight of birds, results obtained from the correlation equations match the computational-fluid-dynamics results more closely than do those obtained from the classical Theodorsen model.

A decomposed Fourier-series solution to Prandtl's classical lifting-line theory is used to examine the effects of rigidbody roll and small-angle wing flapping on the lift, induced-drag, and power coefficients developed by a finite wing. This solution shows that, if the flapping rate for any wing is large enough, the mean induced drag averaged over a complete flapping cycle will be negative, that is, the wing flapping produces net induced thrust. For quasi-steady flapping in pure plunging, the solution predicts that wing flapping has no net effect on the mean lift. A significant advantage of this analytical solution over commonly used numerical methods is the utility provided for optimizing wing-flapping cycles. The analytical solution involves five time-dependent functions that could all be optimized to maximize thrust, propulsive efficiency, and/or other performance measures. Results show that, by optimizing only one of these five functions, propulsive efficiencies exceeding 90% can be attained. For the case of an elliptic planform with linear twist, closed-form relations are presented for the decomposed Fourier coefficients and the flapping rate that produces mean induced thrust that balances the mean drag in the absence of wing flapping.

Towed-water power generators are used in long-distance sailing to generate power for charging battery banks. A basic configuration consists of a spinning turbine (propeller) towed behind the boat and attached to an alternator or generator via a torque line. In the present work, a series of seven inch diameter, six inch pitch turbines were tested both on the water and using computational fluid dynamics techniques. Turbine rotation rates for both an alternator and generator charging a 12V, 80 AH battery were measured for the on-water tests at a nominal towing speed of 3 m/s. Turbine torque vs. rotation rate results obtained by solving the Reynolds-averaged Navier-Stokes equations were plotted against generator and alternator torque curves to predict operating rotations per minute and associated power generation. Predicted rotation rate results were in reasonable agreement with those measured on the water.

Closed-form relations are presented for estimating ratios of the induced-drag and lift coefficients acting on a wing in ground effect to those acting on the same wing outside the influence of ground effect. The closed-form relations for these ground-effect influence ratios were developed by correlating results obtained from numerical solutions to Prandtl's lifting-line theory. Results show that these influence ratios are not unique functions of the ratio of wing height to wingspan, as is sometimes suggested in the literature. These ground-effect influence ratios also depend on the wing planform, aspect ratio, and lift coefficient.

A decomposed Fourier series solution to Prandtl’s classical lifting-line theory is used to predict the lift, induced-thrust, and power coefficients developed by a flapping wing. A significant advantage of this quasi-steady analytical solution over commonly used numerical methods is the utility provided for optimizing wing flapping cycles. The analytical solution involves five time-dependent functions that could all be optimized to maximize thrust, propulsive efficiency, and/or other performance measures. Results show that by optimizing only two of these five functions, propulsive efficiencies exceeding 97% can be obtained. Results are presented for untwisted rectangular wings in pure plunging, rectangular wings with linear washout and the minimum-power washout magnitude, and rectangular wings with the minimum-power washout distribution and magnitude.

Solutions obtained from a numerical method based on Prandtl's lifting-line theory, valid for multiple lifting surfaces with arbitrary sweep, are obtained for a number of rigid wing and sail geometries. The results are compared against solutions obtained using established vortex-lattice methods, and computational fluid dynamics solutions to the Euler equations. For the case of an untwisted, rectangular wing, numerical lifting-line, vortex-lattice, and Euler solutions were all in reasonable agreement. However, the numerical lifting-line method was the only method to predict the constant ratio of induced-drag coefficient to lift coefficient squared, which has been predicted from the analytic solution and confirmed by well established experimental data. Results are also presented for a forward-swept, tapered wing. Additional results are presented in terms of lift and induced-drag coefficients for an isolated mainsail, and mainsail/jib combinations with sails representative of both a standard and tall rig Catalina 27. The influence of the non-linear terms in the lifting-line solution appears minimal, with the exception of mainsail results when considering jib/mainsail combinations. (C) 2013 Elsevier Ltd. All rights reserved.

Based on a more direct analogy between turbulent and molecular transport, a foundation is presented for an energy-vorticity turbulence model. Whereas traditional k-epsilon, k-omega, and k-zeta models relate the eddy viscosity to a dissipation length scale associated with the smaller eddies having the highest strain rates, the proposed model relates the eddy viscosity to a mean vortex wavelength associated with the larger eddies primarily responsible for turbulent transport. A rigorous development of the turbulent-energy-transport equation from the Navier-Stokes equations includes exact relations for the viscous dissipation and molecular transport of turbulent kinetic energy. Application of Boussinesq's analogy between turbulent and molecular transport leads to a transport equation, which shows neither molecular nor turbulent transport of turbulent energy to be simple gradient diffusion. The new turbulent-energy-transport equation contains two closure coefficients: a viscous-dissipation coefficient and a turbulent-transport coefficient. To help evaluate closure coefficients and provide insight into the energy-vorticity turbulence variables, fully rough pipe flow is considered. For this fully developed flow, excellent agreement with experimental data for velocity profiles and friction factors is attained over a wide range of closure coefficients, provided that a given relation between the coefficients is maintained.

A momentum theory which includes the effects of slipstream rotation for wind turbines is presented. The theory accounts for the axial and radial pressure gradients within the slipstream as well as the wake expansion caused by wake rotation. Because of the limiting approximations of previous methods, the effects of slipstream rotation have not been accurately realized. The method included here, which does not suffer from the unrealistic approximations of previous methods, predicts that the effects of slipstream rotation are manifest entirely through an increase in the turbine thrust coefficient. The method predicts, as previous methods do, that the Lanchester–Betz–Joukowski limit of 16/27 is an upper limit for the maximum efficiency, or power coefficient, of a wind turbine. Unlike the results from classical methods that are traditionally reported in terms of the axial induction factor, results of this work are presented in terms of two independent variables, the tip-speed ratio and the torque coefficient. The results included here allow the dependent variables including the thrust coefficient, power coefficient, axial induction factor, and circumferential induction factor to be evaluated in terms of the tip-speed ratio and torque coefficient. Additionally, relationships for the ideal operating conditions of a wind turbine are presented.

Based on a more direct analogy between turbulent and molecular transport, a foundation was recently presented for an energy-vorticity turbulence model. The new turbulent-energytransport equation contains two closure coefficients; a viscous-dissipation coefficient and a turbulent-transport coefficient. To help evaluate the closure coefficients and provide insight into the energy-vorticity turbulence variables, fully rough pipe flow is considered. For this fully developed flow, excellent agreement with experimental data for velocity profiles and friction factors is attained over a wide range of closure coefficients, provided that a given relation between the coefficients is maintained.

Relations are developed for absolute static minimum-control airspeed, without the bank angle restrictions imposed on manned aircraft by regulatory agencies. Absolute static minimum-control airspeed can be important in the design of modern unmanned aerial vehicles, because autonomous computer controlled flight systems do not impose the same constraints as those imposed by human pilots. It is shown that, for a three-channel unmanned aerial vehicle, absolute static minimum-control airspeed always provides the critical constraint for static minimum-control airspeed. Even for the case of a conventional aircraft configuration with both rudder and ailerons, absolute static minimum-control airspeed can be a critical concern for small high-powered propeller-driven unmanned aerial vehicles.

Based on a more direct analogy between turbulent and molecular transport, a foundation is presented for an energy-vorticity turbulence model. Whereas traditional k-e , k-ω , and k-ζ turbulence models relate the eddy viscosity to a dissipation length scale associated with the smaller eddies having the highest strain rates per unit kinetic energy, the proposed model relates the eddy viscosity to a mean vortex wavelength associated with the larger energybearing eddies primarily responsible for turbulent transport. The hypothesized kinematiceddy-viscosity model depends on only the turbulent velocity fluctuations, just as the molecular viscosity depends on only the molecular velocity fluctuations. In contrast, the eddy-viscosity model used in traditional dissipation-based turbulence models results in a kinematic eddy viscosity that is inversely proportional to the molecular viscosity, which violates a fundamental requirement for a Boussinesq model of turbulent transport that is consistent with the definition of the specific Reynolds stress tensor. A rigorous development of the turbulent-energy-transport equation from the Navier-Stokes equations includes exact relations for the viscous dissipation and molecular transport of turbulent kinetic energy. Application of Boussinesq’s analogy between turbulent and molecular transport leads to a transport equation, which shows neither molecular nor turbulent transport of turbulent energy to be simple gradient diffusion.

Solutions obtained from lifting-line, vortex-lattice, and the Euler equations are presented for a series of rigid, thin wing and sail geometries. Initial calculations were performed for an untwisted, rectangular wing. For this case, lifting line theory, vortex lattice, and Euler solutions were all in reasonable agreement. However, the lifting-line theory was the only method to predict a constant ratio of induced drag coefficient to lift coefficient squared. Similar results were found for a forward-swept, tapered wing. Additional results are presented in terms of lift and drag coefficients for an isolated mainsail, and mainsail/jib combinations with sails representative of both a standard and tall rig Catalina 27. Although experimental data is lacking, overall conclusions are that the accuracy realized from lifting-line solutions is as good as or better than that obtained from vortex-lattice solutions and inviscid CFD solutions, but at a fraction of the computational cost. The linear lifting-line results compared quite well with the nonlinear lifting-line results, with the exception of the downstream mainsail when considering jib/mainsail combinations.