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Revisiting the combinatorics of lifting line and 2D vortex lattice theory

Published online by Cambridge University Press:  09 August 2019

M. A. Yukish Open the ORCID record for M. A. Yukish [Opens in a new window]
M. A. Yukish*
Affiliation:
The Applied Research Laboratory The Pennsylvania State UniversityUniversity Park, Pennsylvania, USA
*
may106@psu.edu
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Abstract

This work revisits analyses of lifting line theory by A.R. Collar from 1958, bringing to bear some modern tools and techniques. The 2D vortex lattice model is also considered. Interesting combinatorial properties and simplified expressions for terms are presented, a simplified proof of model convergence shown, extension of the convergence properties to elliptical and arbitrary wing planforms is demonstrated, and approximations provided. An alternative proof of the optimality of constant downwash is presented. Modern automated theorem proving techniques are employed in confirming the combinatorial results.

Keywords

Lifting line modelVortex lattice theoryAutomated theorem proving

Information

Type
Research Article
Information
The Aeronautical Journal , Volume 123 , Issue 1265 , July 2019 , pp. 993 - 1012
Copyright
© Royal Aeronautical Society 2019 

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References

REFERENCES

Collar, A. R. On the accuracy of the representation of a lifting line by a finite set of horseshoe vortices, Aeronautical Quarterly, 1958, 9, (3), pp 232–250.CrossRefGoogle Scholar
Mccormick, B. W. Aerodynamics, Aeronautics, and Flight Mechanics, 2nd ed, Wiley, 1995, New York, xii, pp 652.Google Scholar
Mccormick, B. W. Introduction to Flight Testing and Applied Aerodynamics, Virginia Polytechnic Institute and State University, 2011, Blacksburg, Va.CrossRefGoogle Scholar
Katz, J. and Plotkin, A. Low-Speed Aerodynamics, 2nd ed, Cambridge Aerospace Series, Cambridge University Press, 2001, Cambridge.CrossRefGoogle Scholar
Sugar-Gabor, O. A general numerical unsteady non-linear lifting line model for engineering aerodynamics studies, The Aeronautical Journal, 2018, 122, (1254), pp 1199–1228.CrossRefGoogle Scholar
Phillips, W. F., Hunsaker, D. F. and Joo, J. J. Minimizing induced drag with lift distribution and wingspan, J Aircraft, 2018, 56, (2), pp. 431–441.CrossRefGoogle Scholar
Phillips, W. F. and Hunsaker, D. F. Designing wing twist or planform distributions for specified lift distributions, J Aircraft, 2018, 56, (2), pp. 847–849.CrossRefGoogle Scholar
Collar, A. R. On the reciprocal of a segment of a generalized Hilbert matrix, Mathematical Proceedings of the Cambridge Philosophical Society, 1951, 47, (1), pp. 11–17.CrossRefGoogle Scholar
Deyoung, J. Convergence-proof of discrete-panel wing loading theories, J Aircraft, 1971, 8, (10), pp. 837–839.CrossRefGoogle Scholar
James, R. M. On the remarkable accuracy of the vortex lattice method, Computer Methods in Applied Mechanics and Engineering, 1972, 1, (1), pp. 59–70.CrossRefGoogle Scholar
Branlard, E. Numerical implementation of vortex methods, in Wind Turbine Aerodynamics and Vorticity-Based Methods, Springer, 2017, Cham, Switzerland.CrossRefGoogle Scholar
Bogoya, J. M., Böttcher, A., Grudsky, S. M. and Maximenko, E. A. Maximum norm versions of the Szegõ and Avram–Parter theorems for Toeplitz matrices, J Approximation Theory, 2015, 196, pp. 79–100.CrossRefGoogle Scholar
Böttcher, A., Bogoya, J. M., Grudsky, S. M. and Maximenko, E. A. Asymptotics of eigenvalues and eigenvectors of Toeplitz matrices, Sbornik Mathematics, 2017, 208, (11), pp. 1578–1601.CrossRefGoogle Scholar
Poole, G. and Boullion, T. A survey on M-Matrices, SIAM Review, 1974, 16, (4), pp. 419–427.CrossRefGoogle Scholar
Merikoski, J. On the trace and the sum of elements of a matrix, Linear Algebra and its Applications, 1984, 60, pp. 177–185.CrossRefGoogle Scholar
Johnson, R. Elementary central binomial coefficient estimates, 2014; Available from: https://math.stackexchange.com/q/932509.Google Scholar
Munk, M. M. The Minimum Induced Drag of Aerofoils, National Advisory Committee for Aeronautics, 1923, Washington, DC, United States Germany.Google Scholar
Jones, R. T. The Spanwise Distribution of Lift for Minimum Induced Drag of Wings Having a Given Lift and a Given Bending Moment, National Advisory Committee for Aeronautics. Ames Aeronautical Lab, 1950, Moffett Field, CA, United States.Google Scholar
Fletcher, R. Practical Methods of Optimization, (2nd ed), Wiley-Interscience, 1987, New York.Google Scholar
Koepf, W. Hypergeometric Summation : An Algorithmic Approach to Summation and Special Function Identities, Second edition. Universitat. xvii, Springer, in Heidelberg, 279 pages.Google Scholar
Petkovsek, M., Wilf, H. and Zeilberger, D. A=B, A K Peters, 1996, Wellesley, Massachusetts.CrossRefGoogle Scholar
Paule, P. and Schorn, M.. A Mathematica version of Zeilberger’s algorithm for proving binomial coefficient identities, J Symbolic Computing, 1995, 20, (1), pp. 673–698.CrossRefGoogle Scholar
Harrison, J. Formal proofs of hypergeometric sums, J Automated Reasoning, 2015, 55, (3), pp. 223–243.CrossRefGoogle Scholar
Johnson, C. R. Inverse M-Matrices, Linear Algebra and its Applications, 1982, 47, pp. 195–216.CrossRefGoogle Scholar
Johnson, C. R. and Smith, R. L. Inverse M-Matrices, II, Linear Algebra and its Applications, 2011, 435, (5), pp. 953–983.CrossRefGoogle Scholar
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