Stability of Nonparallel Periodic Flow with Bottom Friction

  • Y. Murakami
  • H. Fukuta
  • K. Gotoh
Part of the International Union of Theoretical and Applied Mechanics book series (IUTAM)

Abstract

Both linear and nonlinear stabilities of a nonparallel periodic flow:ψ = cos y + A cos x, (0 < A < 1) with bottom-friction effect are investigated numerically. We adopt a quasi-two-dimensional (Q2D) approximation which regards the effect of the boundary layer in a vertical direction as a linear damping term proportional to the horizontal velocity in the two-dimensional (2D) Navier-Stokes equation. The linear critical Reynolds number R Lc (λ, A) increases almost linearly as the bottom-friction effect λ increases. R Lc (λ, A) takes the maximum value at A = 1 for any given λ though the energy density of the main flow is an increasing function of A. The nonlinear critical Reynolds number REc(λ, A) by the energy method is an increasing function of λ while it is a decreasing function of A.

Keywords

Vortex Sine Bove 

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References

  1. Bondarenko, N.F. M.Z.Gak,& F.V.Dolzhanskiy,1979 Laboratory and theoretical models of plane periodic flows, Izv. Atmos. Ocean. Phys., 15, 711–716.Google Scholar
  2. Dolzhanskiy, F.V. 1987 Effect of external friction on the stability of plane-parallel flows in a homogeneous incompressible liquid, Izv. Atmos. Ocean. Phys., 23, 262–268.Google Scholar
  3. Dolzhanskiy, F.V. V.A.Krymov,& D. Yu.Manin,1990 Stability and vortex structures of quasi-two-dimensional shear flows, Sov. Phys. Usp, 33, 495–520.ADSCrossRefGoogle Scholar
  4. Gotoh, K. & M.Yamada,1984 Instability in cellular flows, J. Phys. Soc. Jpn, 53, 3395–3398.ADSCrossRefGoogle Scholar
  5. Gotoh, K. & M.Yamada,1986 Instability of the rhombic cell flows, Fluid Dyn. Res., 1, 165–176.ADSCrossRefGoogle Scholar
  6. Matsuda, N. 1993 B.S. in Univ. of Osaka Prefecture, (in Japanese)Google Scholar
  7. Murakami, Y. & H.Fukuta,1993 Nonlinear stability of the Kolmogorov flow with bottom-friction using the energy method, to appear in Proceedings of Analysis of Nonlinear Phenomena and Its Application RIMS, Kyoto.Google Scholar
  8. Murakami, Y. II. Fukuta & K.Gotoh,1993 in preparationGoogle Scholar
  9. Pedlosky, J. 1987 Geophysical Fluid Dynamics, Springer Verlag.MATHCrossRefGoogle Scholar
  10. Sommeria, J. 1986 Experimental study of the two-dimensional inverse energy cascade in a square box, J. Fluid Mech, 170, 139–168.ADSCrossRefGoogle Scholar
  11. Thess, A. 1992a Instabilities in two-dimensional spatially periodic flows. Part I: Kolmogorov flow, Phys. Fluids, A4, 1385–1395.MathSciNetADSGoogle Scholar
  12. Thess, A. 1992b Instabilities in two-dimensional spatially periodic flows. Part II: Square eddy lattice, Phys. Fluids, A4, 1396–1407.MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag, Berlin Heidelberg 1994

Authors and Affiliations

  • Y. Murakami
    • 1
  • H. Fukuta
    • 1
  • K. Gotoh
    • 1
  1. 1.University of Osaka PrefectureSakai 593Japan

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