### Numéro : 1993 - Year : 1985

# Angular criteria for the stability of a helical vortex or a pair of rectilinear vortices - Role of the privileged angles in the optimization of wings sails, hulls in airplanes and ships.

**M. LE RAY, Professeur**

**J. P. DEROYON M. J. DEROYON C. MINAIR,**

*Laboratoire d'hydrodynamique, d'aérodynamique et d'énergétique Université de Valenciennes*

Following the discovery in 1972 by our Laboratory in liquid helium flows, of a discretisation of the angles that an helical vortex can make with its axis, we have progressively shown, as well from existing documents as from visualizations in a wind tunnel or an hydrodynamic tunnel, that this discretization is universally present in the flows of the other fluids, particularly in air flows and water flows, not only in the helical vortices of the marine and air screw propellers, but also in the series of angles existing between the rectilinear vortices present above delta wings. This discretization corresponds to the series of privileged angles with great spacing between two consecutive elements, defined by :

Cos 8 = lï(~+1) , where l and m are both integers, with two families defined by m= ~ (45° ; 35~3 ; 30° ; 26~6 ; 24~1 ; 22~2 ; 20~7 ; 19~4 , 18~4 ;. ..) and m= 2, ~ ~2 (35~3 ; 54~7 ; 63~4 ; 68~6 ; 72° ; 74~5 ; 76~4 ; 77~8 ; 79° ; 80° ;...). The integers ~ and mare similar to these occurring in Legendre polynomials and Spherical Harmonics, which appear in the solutions of numerous three-dimensional problems. It is to be noticed that three of this angles and the relation between them (54~7 = 35~3 + 19~4) are those occurring in the classical theory of the ships wakes shape.

Finally, assimilating the edges and especially the leading edges of the obstacles present in a flow to vortex axes, we have been led to expect and to verify that the stability and optimisation of flows corresponds to the presence of a great amount of privileged angles as well in artificial marine and aerial shapes (wings, sails, cars, hulls, ...) as in some natural shapes.

Very detailed analysis of some of the very numerous structures which have been studied will be presented.

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