Hostname: page-component-848d4c4894-xm8r8 Total loading time: 0 Render date: 2024-06-29T01:59:25.339Z Has data issue: false hasContentIssue false
  • English
  • Français

Minimising induced drag with weight distribution, lift distribution, wingspan, and wing-structure weight

Published online by Cambridge University Press:  19 March 2020

W.F. Phillips ,
D.F. Hunsaker Open the ORCID record for D.F. Hunsaker [Opens in a new window]  and
J.D. Taylor
W.F. Phillips
Affiliation:
Utah State University, Logan, Utah, USA
D.F. Hunsaker*
Affiliation:
Utah State University, Logan, Utah, USA
J.D. Taylor
Affiliation:
Utah State University, Logan, Utah, USA
*
doug.hunsaker@usu.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Because 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.

Keywords

Lifting-line theoryMultidisciplinary design optimisationAerostructural optimisationInduced-drag minimisation
Type
Research Article
Information
The Aeronautical Journal , Volume 124 , Issue 1278 , August 2020 , pp. 1208 - 1235
Copyright
© The Author(s) 2020. Published by Cambridge University Press on behalf of Royal Aeronautical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

This paper was originally presented at the AIAA Aviation Conference in Dallas, Texas and was published in the proceedings of that conference.

References

REFERENCES

Prandtl, L.Trag flügel Theorie, Nachricten von der Gesellschaft der Wissenschaften zu Güttingen, Ges-chäeftliche Mitteilungen, Klasse, 1918, pp 451–477.Google Scholar
Prandtl, L. Applications of modern hydrodynamics to aeronautics, NACA TR-116, June 1921.Google Scholar
Phillips, W.F.Lifting-line analysis for twisted wings and washout-optimized wings, J Aircr, 2004, 41, (1), pp 128–136. (doi:10.2514/1.262)CrossRefGoogle Scholar
Phillips, W.F., Alley, N.R. and Goodrich, W.D.Lifting-line analysis of roll control and variable twist, J Aircr, 2004, 41, (5), pp 1169–1176. (doi:10.2514/1.3846)CrossRefGoogle Scholar
Phillips, W.F.New twist on an old wing theory, Aerospace America, 2005, pp 27–30.Google Scholar
Phillips, W.F., Fugal, S.R. and Spall, R.E.Minimizing induced drag with wing twist, computational-fluid-dynamics validation, J Aircr, 2006, 43, (2), pp 437–444. (doi:10.2514/1.15089)CrossRefGoogle Scholar
Phillips, W.F. and Alley, N.R.Predicting maximum lift coefficient for twisted wings using lifting-line theory, J Aircr, 2007, 44, (3), pp 898–910. (doi:10.2514/1.25640)CrossRefGoogle Scholar
Phillips, W.F.Incompressible Flow over Finite Wings, Mechanics of Flight, 2nd ed., Wiley, 2010, Hoboken, NJ, pp 46–94.Google Scholar
Phillips, W.F., Hunsaker, D.F. and Joo, J.J.Minimizing induced drag with lift distribution and wingspan, J Aircr, 2019, 56, (2), pp 431–441. (doi:10.2514/1.C035027)CrossRefGoogle Scholar
Prandtl, L.Über Tragflügel kleinsten induzierten Widerstandes, Zeitschrift für Flugtechnik und Motorluftschiffahrt, 1933, 24, (11), pp 305–306.Google Scholar
Hunsaker, D.F. and Phillips, W.F. Ludwig Prandtl’s 1933 Paper Concerning Wings for Minimum Induced Drag, Translation and Commentary, AIAA SciTech 2020 Forum, Orlando, Florida, 6–10 January 2020.10.2514/6.2020-0644CrossRefGoogle Scholar
Jones, R.T. The Spanwise Distribution of Lift for Minimum Induced Drag of Wings Having a Given Lift and a Given Bending Moment, NACA TR-2249, December 1950.Google Scholar
Jones, R.T. and Lasinski, T.A. Effect of Winglets on the Induced Drag of Ideal Wing Shapes, NASA TM-81230, September 1980.Google Scholar
Klein, A. and Viswanathan, S.P.Minimum induced drag of wings with given lift and root-bending moment, Zeitschrift fur Angewandte Mathematik und Physik, 1973, 24, pp 886–892.10.1007/BF01590797CrossRefGoogle Scholar
Klein, A. and Viswanathan, S.P.Approximate solution for minimum induced drag of wings with given structural weight, J Aircr, 1975, 12, (2), pp 124–126. (doi:10.2514/3.44425)CrossRefGoogle Scholar
Taylor, J.D. and Hunsaker, D.F. Minimum Induced Drag for Tapered Wings Including Structural Constraints, AIAA-2020-2113, AIAA Scitech 2020 Forum, Orlando, Florida, 6–10 January, 2020. (doi:10.2514/6.2020-2113)CrossRefGoogle Scholar
Lundry, J.L.Minimum swept-wing induced drag with constraints on lift and pitching moment, J Aircr, 1967, 4, pp 73–74. (doi:10.2514/3.43797)CrossRefGoogle Scholar
Lissaman, P.B.S. and Lundry, J.L.A numerical solution for the minimum induced drag of nonplanar wings, J Aircr, 1968, 5, pp 17–21. (doi:10.2514/3.43901)Google Scholar
Ashenberg, J. and Weihsradius, D.Minimum induced drag of wings with curved planform, J Aircr, 1984, 21, pp 89–91. (doi:10.2514/3.56733)CrossRefGoogle Scholar
McGeer, T.Wing design for minimum drag with practical constraints, J Aircr, 1984, 21, pp 879–886. (doi:10.2514/3.45058)CrossRefGoogle Scholar
Rokhsaz, K.Effect of viscous drag on optimum spanwise lift distribution, J Aircr, 1993, 30, pp 152–154. (doi:10.2514/3.46328)CrossRefGoogle Scholar
Demasi, L.Induced drag minimization: a variational approach using the acceleration potential, J Aircr, 2006, 43, pp 669–680. (doi:10.2514/1.15982)CrossRefGoogle Scholar
Demasi, L.Erratum on induced drag minimization: a variational approach using the acceleration potential, J Aircr, 2006, 43, p 1247. (doi:10.2514/1.26648)CrossRefGoogle Scholar
Demasi, L.Investigation on the conditions of minimum induced drag of closed wing systems and C-wings, J Aircr, 2007, 44, pp 81–99. (doi:10.2514/1.21884)CrossRefGoogle Scholar
Pate, D.J. and German, B.J.Lift distributions for minimum induced drag with generalized bending moment constraints, J Aircr, 2013, 50, pp 936–946. (doi:10.2514/1.C032074)CrossRefGoogle Scholar
Demasi, L., Dipace, A., Monegato, G. and Cavallaro, R.Invariant formulation for the minimum induced drag conditions of nonplanar wing systems, AIAA J, 2014, 52, pp 2223–2240. (doi:10.2514/1.J052837)CrossRefGoogle Scholar
Wroblewski, G.E. and Ansell, P.J.Prediction and experimental evaluation of planar wing spanloads for minimum drag, J Aircr, 2017, 54, pp 1664–1674. (doi:10.2514/1.C034156)CrossRefGoogle Scholar
Demasi, L., Monegato, G. and Cavallaro, R.Minimum Induced Drag Theorems for Multiwing Systems, AIAA J, 2017, 55, pp 3266–3287. (doi:10.2514/1.J055652)CrossRefGoogle Scholar
Noll, T.E., Ishmael, S.D., Henwood, B., Perez-Davis, M.E., Tiffany, G.C., Madura, J., Gaier, M., Brown, J.M. and Wierzbanowski, T. Technical Findings, Lessons Learned, and Recommendations Resulting from the Helios Prototype Vehicle Mishap, NATO/RTO AVT-145 Workshop on Design Concepts, Processes, and Criteria for UAV Structural Integrity, Florence, Italy, 14–18 May, 2007.Google Scholar
Vos, R., Gurdal, Z. and Abdalla, M.Mechanism for warp-controlled twist of a morphing wing, J Aircr, 2010, 47, (2), pp 450–457. (doi:10.2514/1.39828)CrossRefGoogle Scholar
Joo, J., Marks, C., Zientarski, L. and Culler, A. Variable camber compliant wing – design, AIAA-2015-1050, 23rd AIAA/AHS Adaptive Structures Conference, Kissimmee, Florida, 5–9 January 2015.CrossRefGoogle Scholar
Marks, C.R., Zientarski, L., Culler, A.J., Hagen, B., Smyers, B.M. and Joo, J.J. Variable camber compliant wing – wind tunnel testing”, AIAA 2015-1051, 23rd AIAA/AHS Adaptive Structures Conference, Kissimmee, Florida, 5–9 January, 2015.CrossRefGoogle Scholar
Miller, S.C., Rumpfkeil, M.P. and Joo, J.J. Fluid-structure interaction of a variable camber compliant wing, AIAA-2015-1235, 53rd AIAA Aerospace Sciences Meeting, Kissimmee, Florida, 5–9 January 2015.CrossRefGoogle Scholar
Joo, J.J., Marks, C.R. and Zientarski, L. Active wing shape reconfiguration using a variable camber compliant wing system, 20th International Conference on Composite Materials, Copenhagen, Denmark, 19–24 July 2015.CrossRefGoogle Scholar
Marks, C.R., Zientarski, L. and Joo, J.J. Investigation into the effect of shape deviation on variable camber compliant wing performance, AIAA-2016-1313, 24th AIAA/AHS Adaptive Structures Conference, San Diego, California, 4–8 January 2016.CrossRefGoogle Scholar
Alley, N.R., Phillips, W.F. and Spall, R.E.Predicting maximum lift coefficient for twisted wings using computational fluid dynamics, J Aircr, 2007, 44, (3), pp 911–917. (doi:10.2514/1.25643)CrossRefGoogle Scholar
Vernengo, G., Bonfiglio, L. and Brizzolara, S.Supercavitating three-dimensional hydrofoil analysis by viscous lifting-line approach, AIAA J, 2017, 55, (12), pp 4127–4141. (doi:10.2514/1.J055504)CrossRefGoogle Scholar
Gamble, L.L., Pankonien, A.M. and Inman, D.J.Stall recovery of a morphing wing via extended nonlinear lifting-line theory, AIAA J, 2017, 55, (9), pp 2956–2963. (doi:10.2514/1.J055042)CrossRefGoogle Scholar
Izraelevitz, J.S., Zhu, Q. and Triantafyllou, M.S.State-space adaptation of unsteady lifting line theory: twisting/flapping wings of finite span, AIAA J, 2017, 55, (4), pp 1279–1294. (doi:10.2514/1.J055144)CrossRefGoogle Scholar
Gallay, S. and Laurendeau, E.Preliminary-design aerodynamic model for complex configurations using lifting-line coupling algorithm, J Aircr, 2016, 53, (4), pp 1145–1159. (doi:10.2514/1.C033460)CrossRefGoogle Scholar
Gallay, S. and Laurendeau, E.Nonlinear generalized lifting-line coupling algorithms for pre/poststall flows, AIAA J, 2015, 53, (7), pp 1784–1792. (doi:10.2514/1.J053530)CrossRefGoogle Scholar
Spalart, P.R.Prediction of lift cells for stalling wings by lifting-line theory, AIAA J, 2014, 52, (8), pp 1817–1821. (doi:10.2514/1.J053135)CrossRefGoogle Scholar
Phillips, W.F.Analytical decomposition of wing roll and flapping using lifting-line theory, J Aircr, 2014, 51, (3), pp 761–778. (doi:10.2514/1.C032399)CrossRefGoogle Scholar
Phillips, W.F. and Hunsaker, D.F.Lifting-line predictions for induced drag and lift in ground effect, J Aircr, 2013, 50, (4), pp 1226–1233. (doi:10.2514/1.C032152)CrossRefGoogle Scholar
Wickenheiser, A.M. and Garcia, E.Extended nonlinear lifting-line method for aerodynamic modeling of reconfigurable aircraft, J Aircr, 2011, 48, (5), pp 1812–1817. (doi:10.2514/1.C031406)CrossRefGoogle Scholar
Junge, T., Gerhardt, F.C., Richards, P. and Flay, R.G.J.Optimizing spanwise lift distributions yacht sails using extended lifting line analysis, J Aircr, 2010, 47, (6), pp 2119–2129. (doi:10.2514/1.C001011)CrossRefGoogle Scholar
Wickenheiser, A. and Garcia, E.Aerodynamic modeling of morphing wings using an extended lifting-line analysis, J Aircr, 2007, 44, (1), pp 10–16. (doi:10.2514/1.18323)CrossRefGoogle Scholar
Sugimoto, T.Induced velocity in the plane of an elliptically loaded lifting line, AIAA J, 2002, 40, (6), pp 1233–1236. (doi:10.2514/2.1776)CrossRefGoogle Scholar
Phillips, W.F. and Snyder, D.O.Modern adaptation of prandtl’s classic lifting-line theory, J Aircr, 2000, 37, (4), pp 662–670. (doi:10.2514/2.2649)CrossRefGoogle Scholar
Rasmussen, M.L. and Smith, D.E.Lifting-line theory for arbitrarily shaped wings, J Aircr, 1999, 36, (2), pp 340–348. (doi:10.2514/2.2463)CrossRefGoogle Scholar
Iosilevskii, G.Lifting-line theory of an arched wing in asymmetric flight, J Aircr, 1996, 33, (5), pp 1023–1026. (doi:10.2514/3.47050)CrossRefGoogle Scholar
Jadic, I. and Constantinescu, V.N.Lifting line theory for supersonic flow applications, AIAA J, 1993, 31, (6), pp 987–994. (doi:10.2514/3.11718)CrossRefGoogle Scholar
Plotkin, A. and Tan, C.H.Lifting-line solution for a symmetrical thin wing in ground effect, AIAA J, 1986, 24, (7), pp 1193–1194. (doi:10.2514/3.9413)Google Scholar
Cheng, H.K., Meng, S.Y., Chow, R. and Smith, R.C.Transonic swept wings studied by the lifting-line theory, AIAA J, 1981, 19, (8), pp 961–968. (doi:10.2514/3.7837)CrossRefGoogle Scholar
Anderson, J.D. and Corda, S.Numerical lifting line theory applied to drooped leading-edge wings below and above stall, J Aircr, 1980, 17, (12), pp 898–904. (doi:10.2514/3.44690)CrossRefGoogle Scholar
Cheng, H.K. and Meng, S.Y.Lifting-line theory of oblique wings in transonic flows, AIAA J, 1979, 17, (1), pp 121–124. (doi:10.2514/3.61081)CrossRefGoogle Scholar
Cheng, H.K.Lifting-line theory of oblique wings, AIAA J, 1978, 16, (11), pp 1211–1213. (doi:10.2514/3.61033)CrossRefGoogle Scholar
Small, R.D.Transonic lifting line theory - Numerical procedure for shock-free flows, AIAA J, 1978, 16, (6), pp 632–634. (doi:10.2514/3.7562)CrossRefGoogle Scholar
Lan, C.E. and Fasce, M.H.Applications of an improved nonlinear lifting-line theory, J Aircr, 1977, 14, (4), pp 404–407. (doi:10.2514/3.44601)CrossRefGoogle Scholar
Bera, R.K.Comment on “Solution of the Lifting Line Equation for Twisted Elliptic Wings”, J Aircr, 1975,12, (6), pp 561–562. (doi:10.2514/3.59834)CrossRefGoogle Scholar
Bera, R.K.Some remarks on the solution of the lifting line equation, J Aircr, 1974, 11, (10), pp 647–648. (doi:10.2514/3.44397)CrossRefGoogle Scholar
Lan, C.T.An improved nonlinear lifting-line theory, AIAA J, 1973, 11, (5), pp 739–742. (doi:10.2514/3.6819)CrossRefGoogle Scholar
Kerney, K.P.A correction to ‘lifting-line theory as a singular perturbation problem’, AIAA J, 1972, 10, (12), pp 1683–1684. (doi:10.2514/3.6702)CrossRefGoogle Scholar
Filotas, L.T.Solution of the lifting line equation for twisted elliptic wings, J Aircr, 1971, 8, (10), pp 835–836. (doi:10.2514/3.44308).CrossRefGoogle Scholar
Lakshminarayana, B.Extension of lifting-line theory to a cascade of split aerofoils, AIAA J, 1964, 2, (5), pp 938–940. (doi:10.2514/3.2461)CrossRefGoogle Scholar
Anderson, R.C. and Millsaps, K.Application of the galerkin method to the prandtl lifting line equation, J Aircr, 1964, 1, (3), pp 126–128. (doi:10.2514/3.43565)CrossRefGoogle Scholar
Tuyl, A.V.The replacement of lifting surfaces by lifting lines with variable position, J Aerospace Sciences, 1959, 26, (2), pp 127–128. (doi:10.2514/8.7967)CrossRefGoogle Scholar
Dengler, M.A.Concerning “The subsonic calculation of circulatory spanwise loadings for oscillating airfoils by lifting-line techniques”, J Aeronautical Sciences, 1953, 20, (6), pp 439–439. (doi:10.2514/8.2673)CrossRefGoogle Scholar
Dengler, M.A.The subsonic calculation of circulatory spanwise loadings for oscillating airfoils by lifting-line techniques, J Aeronautical Sciences, 1952, 19, (11), pp 751–759. (doi:10.2514/8.2459)CrossRefGoogle Scholar
Reissner, E.Note on the relation of lifting-line theory to lifting-surface theory, J Aeronautical Sciences, 1951, 18, (3), pp 212–214. (doi:10.2514/8.1905)CrossRefGoogle Scholar
Flax, A.H.On a variational principle in lifting-line theory, J Aeronautical Sciences, 1950, 17, (9), pp 596–597. (doi:10.2514/8.1732)CrossRefGoogle Scholar
Thomas, F.Appendix 1: Sailplane Design Data and Drawings, Fundamentals of Sailplane Design, Translated by Milgram, J., College Park Press, College Park, MD, 1999, pp 195–244.Google Scholar
Stewart, A.J. and Hunsaker, D.F. Minimization of Induced and Parasitic Drag on Variable-Camber Morphing Wings, AIAA-2020-0277, AIAA Scitech 2020 Forum, Orlando, Florida, 6–10 January, 2020. (doi:10.2514/6.2020-0277)CrossRefGoogle Scholar
Boeing 777 Airplane Rescue and Fire Fighting Information, Boeing Commercial Airplanes, October, 2018.Google Scholar
777-200/300 Airplane Characteristics for Airport Planning, D6-58329, Boeing Commercial Airplanes, July, 1998.Google Scholar
Langton, R., Clark, C., Hewitt, M. and Richards, L.Fuel System Functions of Commercial Aircraft, Aircraft Fuel Systems, Wiley, Hoboken, NJ, 2009, pp 53–95. (doi: 10.1002/9780470059470)CrossRefGoogle Scholar