Exact bounds for lift-to-drag ratios of profiles in the Helmholtz-Kirchhoff flow. (English) Zbl 1309.76116
Summary: 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 \(C_D\) under a given lift coefficient \(C_L\). 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 \(C_L\) and \(C_D\) and define maximum and minimum values of the lift-to-drag ratios \(C_L/C_D\) for a given \(C_L\).
MSC:
| 76G25 | General aerodynamics and subsonic flows |
References:
| [1] | DOI: 10.1017/S0022112098004017 · Zbl 0923.76235 · doi:10.1017/S0022112098004017 |
| [2] | Sov. Phys., Dokl. 33 pp 11– (1988) |
| [3] | Course of the Calculus of Variations (1950) |
| [4] | Journal für die reine und angewandte Mathematik 70 pp 289– (1869) |
| [5] | Monatsberichte der Königlichen Akademie der Wissenschaften zu Berlin 23 pp 215– (1868) |
| [6] | Inequalities (1934) · Zbl 0010.10703 |
| [7] | DOI: 10.1016/j.cam.2003.10.006 · Zbl 1220.76023 · doi:10.1016/j.cam.2003.10.006 |
| [8] | Problems of Shape Optimization in Aerohydrodynamics (2008) |
| [9] | Eur. J. Appl. Maths. 6 pp 517– (1995) |
| [10] | Mathematical Methods of Airofoil Design. Inverse Boundary-Value Problems of Aerohydrodynamics (1997) |
| [11] | DOI: 10.1017/jfm.2011.358 · Zbl 1241.76113 · doi:10.1017/jfm.2011.358 |
| [12] | DOI: 10.1134/S1028335811090096 · doi:10.1134/S1028335811090096 |
| [13] | DOI: 10.1017/S0022112005005835 · Zbl 1082.76024 · doi:10.1017/S0022112005005835 |
| [14] | The Theory of Jets in an Ideal Fluid (1966) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
