# zbMATH — the first resource for mathematics

Force-enhancing vortex equilibria for two parallel plates in uniform flow. (English) Zbl 1364.76023
Summary: A two-dimensional potential flow in an unbounded domain with two parallel plates is considered. We examine whether two free point vortices can be trapped near the two plates in the presence of a uniform flow and observe whether these stationary point vortices enhance the force on the plates. The present study is an extension of previously published work in which a free point vortex over a single plate is investigated. The flow problem is motivated by an airfoil design problem for the double wings. Moreover, it also contributes to a design problem for an efficient wind turbine with vertical blades. In order to obtain the point-vortex equilibria numerically, we make use of a linear algebraic algorithm combined with a stochastic process, called the Brownian ratchet scheme. The ratchet scheme allows us to capture a family of stationary point vortices in multiply connected domains with ease. As a result, we find that stationary point vortices exist around the two plates and they enhance the downward force and the counter-clockwise rotational force acting on the two plates.

##### MSC:
 76B10 Jets and cavities, cavitation, free-streamline theory, water-entry problems, airfoil and hydrofoil theory, sloshing 76D17 Viscous vortex flows 76B47 Vortex flows for incompressible inviscid fluids 76G25 General aerodynamics and subsonic flows
Full Text:
##### References:
 [1] Crowdy, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 462 (2069) pp 1387– (2006) · Zbl 1149.76634 · doi:10.1098/rspa.2005.1631 [2] THEOR COMPUT FLUID DYN 24 pp 9– (2010) · Zbl 1191.76020 · doi:10.1007/s00162-009-0098-5 [3] Crowdy, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 461 (2060) pp 2477– (2005) · Zbl 1186.76630 · doi:10.1098/rspa.2005.1492 [4] COMPUT METHODS FUNCT THEORY 6 pp 59– (2006) · Zbl 1101.30010 · doi:10.1007/BF03321118 [5] AIAA J 20 pp 292– (1982) · Zbl 0479.76018 · doi:10.2514/3.7913 [6] AGARD CONF PROC 247 pp 15– (1978) [7] J APPL MECH 63 pp 543– (1996) · Zbl 0885.76014 · doi:10.1115/1.2788902 [8] Newton, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 463 (2082) pp 1525– (2007) · Zbl 1347.76014 · doi:10.1098/rspa.2007.1832 [9] 15 pp 618– (1978) · doi:10.2514/3.58416 [10] 57 pp 107– (1977) · Zbl 0385.76033 · doi:10.1002/sapm1977572107 [11] Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 465 (2108) pp 2589– (2009) · Zbl 1186.37100 · doi:10.1098/rspa.2009.0070 [12] SOARING 38 pp 26– (1974) [13] J FLUID MECH 562 pp 151– (2006) · Zbl 1157.76316 · doi:10.1017/S0022112006001054
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.