Abstract
We consider Kolmogorov's problem of viscous incompressible fluid motions on two dimensional tori. The problem is a bifurcation problem with two parameters, the Reynolds number and the aspect ratio. Varying the aspect ratio as a splitting parameter, we compute numerically bifurcation diagrams with the Reynolds number as a bifurcation parameter. As the aspect ratio changes, we observe turning points and secondary bifurcation points appear or disappear. Furthermore, Hopf bifurcation points are also found when the aspect ratio of the torus satisfies a certain condition. This paper is an improved and enlarged version of the report [26]. Some errors in [17, 26] are corrected here.
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References
A.L. Afendikov and K.I. Babenko, Bifurcation in the presence of a symmetry group and loss of stability of some plane flows of a viscous fluid. Soviet Math. Dokl.,33 (1986), 742–747.
K.I. Babenko and M.M. Vasil'ev, Stability of incompressible fluid flows on a plane torus. Soviet Phys. Dokl.,32 (1987), 541–543.
D.N. Beaumont, The stability of spatially periodic flows. J. Fluid. Mech.,108 (1981), 461–474.
G.K. Batchelor, On steady laminar flow with closed streamlines at large Reynolds number. J. Fluid Mech.,1 (1956), 177–190.
S.O. Belotserkovskii, A.P. Mirabel' and M.A. Chusov, On the construction of the post-critical mode for a plane periodic flow. Izv. Atmos. Oceanic Phys.,14 (1978), 6–12.
N.F. Bondarenko, M.Z. Gak and F.V. Dolzhanskii, Laboratory and theoretical models of plane periodic flows. Izv. Atmos. Oceanic Phys.,15 (1979), 711–716.
V. Franceschini, C. Tebaldi and F. Zironi, Fixed point limit behavior ofN-mode truncated Navier-Stokes equations asN increases. J. Statist. Phys.,35 (1984), 387–397.
R. Kobayashi, D. Takahashi and H. Nakano, GLSC: the Graphics Library for Scientific Computing. Ryukoku University.
K. Gotoh and M. Yamada, Instability of a cellular flow. J. Phys. Soc. Japan,53 (1984), 3395–3398.
K. Gotoh and M. Yamada, Stability of spatially periodic flows. Chapter 19 of Encyclopedia of Fluid Mechanics, Gulf Publishing Company, Houston TX, U.S.A., 1986.
K. Gotoh and M. Yamada, The instability of rhombic cell flows. Fluid Dynam. Res.,1 (1986), 165–176.
K. Gotoh, M. Yamada and J. Mizushima, The theory of stability of spatially periodic parallel flows. J. Fluid Mech.,127 (1983), 45–58.
R. Grappin, J. Leorat and P. Londrillo, Onset and development of turbulence in two dimensional periodic shear flows. J. Fluid Mech.,195 (1988), 239–256.
J.S.A. Green, Two-dimensional turbulence near the viscous limit. J. Fluid Mech.,62 (1974), 273–287.
V.I. Iudovich, Example of the generation of a secondary stationary or periodic flow when there is loss of stability of the laminar flow of a viscous incompressible fluid. J. Appl. Math. Mech.,29 (1965), 527–544.
T. Kambe and Y. Kamanaka, Nonlinear dynamics of two-dimensional flows: bifurcation, chaos and turbulence. To appear in Proc. Nonlinear Dynamics of Structures (Moscow-Permi Symposium), World Scientific.
M. Katsurada, H. Okamoto and M. Shōji, Bifurcation of the stationary Navier-Stokes flows on a 2-D torus. To appear in Proc. Katata Workshop on Nonlinear PDEs and Applications (eds. M. Mimura and T. Nishida).
H.B. Keller, Lectures on Numerical Methods in Bifurcation Theory. Tata Institute of Fundamental Research No. 79, Springer Verlag, 1987.
S. Kida, M. Yamada and K. Ohkitani, The energy spectrum in the universal range of two dimensional turbulence. Fluid Dynam. Res.,4 (1988), 271–301.
S. Kida, K. Ohkitani and M. Yamada, Triply periodic motion in a Navier-Stokes flow. J. Phys. Soc. Japan,58 (1989), 2380–2385.
C. Marchioro, An example of absence of turbulence for any Reynolds number. Comm. Math. Phys.,105 (1986), 99–106.
L.D. Meshalkin and Y.G. Sinai, Investigation of the stability of a stationary solution of a system of equations for the plane movement of an incompressible viscous liquid. J. Appl. Math. Mech.,25 (1962), 1700–1705.
B. Nicolaenko and Z.-S. She, Coherent structures, homoclinic cycles and vorticity explosions in Navier-Stokes flows. Topological Fluid Mechanics (eds. H.K. Moffatt and A. Tsinober), 1990, 265–277.
A.M. Obukhov, Kolmogorov flow and laboratory simulation of it. Russian Math. Surveys,38:4 (1983), 113–126.
A.M. Obukhov, F.V. Dolzhanskii and A.M. Batchaev, A simple hydrodynamic system with oscillating topology. Topological Fluid Mechanics (eds. H.K. Moffatt and A. Tsinober), 1990, 304–314.
H. Okamoto and M. Shōji, Bifurcation diagrams in Kolmogorov's problem of viscous incompressible fluid on 2-D flat tori. Kyoto Univ. Res. Inst. Math. Sci. Kokyuroku,767 (1991), 29–51.
G.I. Sivashinsky, Weak turbulence in periodic flows. Physica,17D (1985), 243–255.
M. Takaoka, Stability of triangular cell flows. J. Phys. Soc. Japan,58 (1989), 2223–2226.
C. Tebaldi, Fixed points behaviour in truncated two-dimensional Navier-Stokes equations. Nonlinear Dynamics (ed. G. Turchetti), World Scientific, 1989, 412–422.
M. Yamada, Nonlinear stability theory of spatially periodic parallel flows. J. Phys. Soc. Japan,55 (1986), 3073–3079.
Y. Yan, To appear.
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Okamoto, H., Shōji, M. Bifurcation diagrams in Kolmogorov's problem of viscous incompressible fluid on 2-D flat tori. Japan J. Indust. Appl. Math. 10, 191 (1993). https://doi.org/10.1007/BF03167572
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DOI: https://doi.org/10.1007/BF03167572