[Spaces of universal flows, 3.]
Universal motions with uniform steady vorticity form a corolla of linear spaces derived from rigid body motions. Closely related to potential flows, they satisfy two extensions of Lagrange theorem, are investigated with the help of complex functions, as stand celebrated when be plane. They take place in hydrodynamics, aerodynamics, geophysics, astrophysics, turbulence, physics of plasmas and superfluid helium. In all the cases, arbitrary unsteady span-wise translations permit to generalise as well as to exhibit helical or 3D universal motions. Three misunderstood periodic flows illustrate our purpose, as they approach shear instabilities in numerous fluids.
Les mouvements premiers à vorticité stationnaire uniforme composent une corolle d'espaces vectoriels isomorphes dérivés des déplacements solides ou des écoulements de Couette. Proches cadets des écoulements potentiels, ils vérifient deux extensions du théorème de Lagrange, s'étudient à l'aide des fonctions holomorphes, et sont célèbres quand ils sont plans. Ils interviennent en hydrodynamique, aérodynamique, géophysique, astrophysique, turbulence, physiques des plasmas et de l'hélium superfluide. À chaque fois, des translations transversales non stationnaires arbitraires permettent de générer des mouvements premiers tridimensionnels complexes. Trois écoulements périodiques méconnus servent d'illustration, et approchent l'instabilité de cisaillement dans pléthore de fluides.
Accepted:
Published online:
Keywords: Material engineering, Rankine, Kirchhoff and Batchelor's vortex patches, Visco-elastic critical layers, Tollmien–Schlichting waves
Michel Bouthier 1
@article{CRMECA_2004__332_12_1019_0,
author = {Michel Bouthier},
title = {Espaces d'\'ecoulements dits {\guillemotleft} universels {\guillemotright}, 3},
journal = {Comptes Rendus. M\'ecanique},
pages = {1019--1025},
publisher = {Elsevier},
volume = {332},
number = {12},
year = {2004},
doi = {10.1016/j.crme.2004.05.004},
language = {fr},
}
Michel Bouthier. Espaces d'écoulements dits « universels », 3. Comptes Rendus. Mécanique, Volume 332 (2004) no. 12, pp. 1019-1025. doi : 10.1016/j.crme.2004.05.004. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2004.05.004/
[1] Espaces d'écoulements dits « universels », Part 2, C. R. Mécanique, Volume 331 (2003), pp. 165-172 Section 4, Éq. (12) ; Section 1, Eq. (3) Section 1
[2] Stability of the Rankine vortex in a multipolar strain field, Phys. Fluids, Volume 13 (2001) no. 3, pp. 660-676 Eq. (1.1a)
[3] Emergence and evolution of triangular vortices, J. Fluid Mech., Volume 259 (1994), pp. 305-331
[4] Symmetrical Joukowsky airfoils in shear flow, Quart. Appl. Math., Volume 1 (1943), pp. 130-148
[5] Exact solutions of the Navier–Stokes equations – the generalized Beltrami flows, review and extension, Acta Mech., Volume 81 (1990), pp. 69-74 Source or vortex in shear flow, Eqs. (12)–(13); Shear flow over convection cells, Eq. (14)
[6] Theoretical Aerodynamics, 1966 (reedited by Dover, New York, 1973; Section 5-72, Uniform shear flow; Section 5-73, Circular cylinder in uniform shear flow; pp. 93–95; Section 13-13, Rankine's combined vortex, pp. 355–356; Section 13-71, Single infinite row, p. 375; Examples XIII 24, 26, p. 389)
[7] Sur les écoulements de cisaillement à tourbillon constant autour d'un profil animé d'un mouvement quelconque, J. Mec., Volume 13 (1974), pp. 1-30
[8] Counter-rotating vortex patches in shear: a model of the effect of wind shear on aircraft trailing vortices, Proc. R. Soc. London Ser. A, Volume 458 (2002), pp. 1527-1553
[9] An Introduction to Fluid Dynamics, Cambridge Math. Lib., Cambridge University Press, Cambridge, UK, 2000 (Section 7.4, Steady two-dimensional flow with vorticity throughout the fluid, pp. 538–543)
[10] The Theory of Rotating Fluids, Cambridge University Press, Cambridge, UK, 1968 (Section 1.1, A few experiments, pp. 1–5)
[11] Vortex Dynamics, Dover, 1997 (Chapter 9, Dynamics of two-dimensional vortex patches, pp. 160–191)
[12] Théorie des tourbillons, Gauthiers, Villars, Paris, 1930 (Chapiter VIII, Tourbillons de dimensions finies. Le tourbillon elliptique de Kirchhoff, pp. 187–198)
[13] A proposal concerning laminar wakes behind bluff bodies at large Reynolds number, J. Fluid Mech., Volume 1 (1956), pp. 388-398
[14] The calculation of some Batchelor flows: The Sadovskii vortex and rotational corner flow, Phys. Fluids, Volume 31 (1988) no. 5, pp. 978-990
[15] Multipolar vortices in two-dimensional incompressible flows, J. Fluid Mech., Volume 267 (1994), pp. 23-51
[16] Multipolar vortices and algebraic curves, Proc. R. Soc. London Ser. A, Volume 457 (2001), pp. 2337-2359
[17] Motion of an elliptic vortex in a uniform shear flow, J. Phys. Soc. Japan, Volume 50 (1981), pp. 3517-3520
[18] Vortex crystals from 2D Euler flow: experimentation and simulation, Phys. Fluids, Volume 11 (1999), pp. 905-914
[19] Point vortex dynamics within a background vorticity patch, Phys. Fluids, Volume 13 (2001) no. 3, pp. 677-691
[20] Exact solutions for rotating vortex arrays with finite-area cores, J. Fluid Mech., Volume 469 (2002), pp. 209-235
[21] The dynamics of a columnar vortex in an imposed strain, Phys. Fluids, Volume 27 (1984), pp. 2397-2402
[22] Stability analysis of a two-dimensional vortex pattern, Phys. Rev. E, Volume 55 (1997) no. 6, pp. 7010-7014
[23] Hydrodynamic Stability, Cambridge University Press, Cambridge, UK, 1981 (Fig. 4.3, p. 141)
[24] Critical layers in shear flows, Annu. Rev. Fluid Mech., Volume 18 (1986), pp. 405-432 (Fig. 2, p. 411)
[25] Shear flow over convection cells-an exact solution of the Navier–Stokes equations, Z. Angew. Math. Mech., Volume 70 (1990) no. 8, pp. 351-352
Cited by Sources:
Comments - Policy