Vortices of large scale appearing in the 2D stationary Navier–Stokes equations at large Reynolds numbers
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We consider Kolmogorov’s problem for the two-dimensional (2D) Navier–Stokes equations. Stability of and bifurcation from the trivial solution are studied numerically. More specifically, we compute solutions with large Reynolds numbers with a family of prescribed external forces of increasing degree of oscillation. We find that, whatever the external force may be, a stable steady-state of simple geometric character exits for sufficiently large Reynolds numbers. We thus observe a kind of universal outlook of the solutions, which is independent of the external force. This observation is reinforced further by an asymptotic analysis of a simple equation called the Proudman–Johnson equation.
KeywordsNavier–Stokes equations Subharmonic bifurcation Proudman–Johnson equation Vortex of large scale
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- 7.Drazin P.G., Reid W.H.: Hydrodynamic Stability, 2nd edn. Cambridge University Press, Cambridge (2004)Google Scholar
- 10.Govaerts, W.J.F.: Numerical methods for bifurcation of dynamical equilibria. SIAM (2000)Google Scholar
- 11.Heywood, J.G., Nagata, W., Xie, W.: A numerically based existence theorem for the Navier–Stokes equations. J. Math. Fluid. Mech., pp. 5–23 (1999)Google Scholar
- 14.Jones W.B., Thron W.J.: Continued Fractions: Analytic Theory and Applications. Addison- Wesley, Reading (1980)Google Scholar
- 24.Okamoto H.: A study of bifurcation of Kolmogorov flows with an emphasis on the singular limit. Docum. Math., Proc. Int. Congress Math. III, 523–532 (1998)Google Scholar
- 27.Wall, H.S.: Analytic Theory of Continued Fractions. Van Nostrand, New York (1948), reprint by Amer. Math. Soc. (2000)Google Scholar