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Exact bounds for lift-to-drag ratios of profiles in the Helmholtz–Kirchhoff flow

Published online by Cambridge University Press:  17 January 2014

D. V. MAKLAKOV  and
I. R. KAYUMOV
D. V. MAKLAKOV
Affiliation:
Lobachevsky Institute of Mathematics and Mechanics, Kazan Federal (Volga Region) University, Kremlyovskaya, 35, Kazan 420008, Russia email: [email protected], [email protected]
I. R. KAYUMOV
Affiliation:
Lobachevsky Institute of Mathematics and Mechanics, Kazan Federal (Volga Region) University, Kremlyovskaya, 35, Kazan 420008, Russia email: [email protected], [email protected]
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Abstract

In this work we investigate limiting values of the lift and drag coefficients of profiles in the Helmholtz–Kirchhoff (infinite cavity) flow. The coefficients are based on the wetted arc length of profile surfaces. The problem is to find global minimum and maximum values of the drag coefficient CD under a given lift coefficient CL. We reduce the problem to a constrained problem of calculus of variations and solve it analytically. In so doing we do not only determine extremals but also strictly prove that these extremals realize global extrema. The proofs are based on non-trivial application of Jensen's inequality. The solution of the problem allows us to construct the domain of possible variations of coefficients CL and CD and define maximum and minimum values of the lift-to-drag ratios CL/CD for a given CL.

Keywords

Extremal problemIdeal fluidPotential flowsHelmholtz–Kirchhoff modelCavity flowsLift-to-drag ratio
Type
Papers
Information
European Journal of Applied Mathematics , Volume 25 , Issue 2 , April 2014 , pp. 231 - 254
Copyright
Copyright © Cambridge University Press 2014 

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References

[1]Elizarov, A. M., Il'inskiy, N. B. & Potashev, A. V. (1997) Mathematical Methods of Airofoil Design. Inverse Boundary-Value Problems of Aerohydrodynamics, Akademy Verlag, Berlin, Germany.Google Scholar
[2]Elizarov, A. M., Kasimov, A. R. & Maklakov, D. V. (2008) Problems of Shape Optimization in Aerohydrodynamics, Fizmatlit, Moscow, Russia. (in Russian)Google Scholar
[3]Gurevich, M. I. (1966) The Theory of Jets in an Ideal Fluid, Pergamon Press, Oxford, UK.Google Scholar
[4]Hardy, G., Littlewood, J. & Polya, G. (1934) Inequalities, Cambridge University Press, Cambridge, UK.Google Scholar
[5]Helmholtz, H. L. F. (1868) Über discontinuierliche Flüssigkeitsbewegungen. Monatsberichte der Königlichen Akademie der Wissenschaften zu Berlin 23, 215–228 (reprinted in 1868 in Philosophical Magazine 36, 337–346.Google Scholar
[6]Kirchhoff, G. (1869) Zur Theorie freier Flüssigkeitsstrahlen Journal für die reine und angewandte Mathematik 70, 289–298.Google Scholar
[7]Lavrentyev, M. A. & Lusternik, L. A. (1950) Course of the Calculus of Variations, Gostekhteorizdat, Moscow, Russia. (in Russian)Google Scholar
[8]Maklakov, D. V. (1988) The maximum resistance of a curvilinear obstacle subjected to the action of free jets with separation. Sov. Phys., Dokl. 33 (1), 11–13; translation from 1988 Dokl. Akad. Nauk SSSR 298(3), 574–577.Google Scholar
[9]Maklakov, D. V. (1999) A note on the optimum profile of a spray less planning surface. J. Fluid Mech. 384, 281–292.Google Scholar
[10]Maklakov, D. V. (2004) Some remarks on the exact solution for an optimal impermeable parachute problem. J. Comput. Appl. Maths. 166 (2), 591–596.Google Scholar
[11]Maklakov, D. V. (2005) On deflectors of optimum shape. J. Fluid Mech. 540, 175–187.CrossRefGoogle Scholar
[12]Maklakov, D. V. (2011) Analog of the Kutta–Joukowskii theorem for the Helmholtz–Kirchhoff flow past a profile. Dokl. Phys. 56 (11), 573–576; translation from Dokl. Akad. Nauk 441(2), 187–190.CrossRefGoogle Scholar
[13]Maklakov, D. V. (2011) On the lift and drag of cavitating profiles and the maximum lift and drag. J. Fluid Mech. 687, 360–375.Google Scholar
[14]Maklakov, D. V. & Uglov, A. N. (1995) On the maximum drag of a curved plate in flow with a wake. Eur. J. Appl. Maths. 6 (5), 517–527.CrossRefGoogle Scholar