Abstract
We review recent work on the transition to turbulence in shear flows, in particular plane Couette flow. In the linearized system the non-normality of the linear operator gives rise to non-orthogonal eigenvectors and to a significant amplification of noise. We present a typical result for a Fokker-Planck equation with non-normal relaxation matrix. As the driving becomes stronger, further stationary states are born in a saddle node bifurcation. This is observed in a two degree of freedom phenomenological model, in the a few mode approximation to a simple shear flow and in full numerical studies of plane Couette flow. The stationary states give rise to a fractal border between decaying and turbulent states.
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References
Ahydin, M. and Leutheusser, H.J. (1979): Novel experimental facility for the study of plane Couette flow. Rev. Sci. Instrum. 50, 1362–1366
Baggett, J.S. and Trefethen, L.N. (1997): Low-dimensional models of subcritical transition to turbulence, Phys. Fluids 9, 1043–1053
Boberg, L. and Brosa, U. (1988): Onset of turbulence in a pipe, Z. Naturforsch. 43a, 697
Chandrasekhar, S. (1961): Hydrodynamic and hydromagnetic stability, Oxford University Press
Clever, R.M. and Busse, F.H. (1997): Tertiary and quarternary solutions in plane Couette flow, J. Fluid Mech. 344, 137
Cross, M.C. and Hohenberg, P.C. (1993): Pattern formation outside equilibrium, Rev. Mod. Phys. 65, 851
Darbyshire, A.G. and Mullin, T. (1995): Transition to turbulence in constant-mass-flux pipe flow, J. Fluid Mech. 289, 83–114
Dauchot, O. and Daviaud F. (1995): Streamwise vortices in plane Couette flow, Phys. Fluid. 7, 901–903
Daviaud F., Hegseth J. and Bergé P. (1992): Subcritical transition to turbulence in plane Couette flow, Phys. Rev. Lett. 69, 2511–2514
Drazin, P.G. and Reid, W.H. (1981): Hydrodynamic stability, Cambridge University Press
Eckhardt, B. (1988): Irregular scattering, Physica D 33, 89
Eckhardt, B. and Aref, H. (1988): Integrable and chaotic motion of four vortices: II collision dynamics of vortx pairs, Phil. Trans. R. Soc. London A 326, 655
Eckhardt, B. and Mersmann, A. (1997): Transition to turbulence in a shear flow, preprint
Grossmann, S. (1996): in Nonlinear physics of complex systems, J. Parisi, S.C. Müller and W. Zimmermann, Springer, Berlin, pp. 10
Gebhardt, T. and Grossmann, S. (1994): Chaos transition despite linear stability, Phys. Rev. E 50, 3705
Lorenz, E.N. (1963): Deterministic nonperiodic flow, J. Atmos. Sci. 23, 130
Lundbladh, A. and Johansson A.V. (1991): Direct simulation of turbulent spots in plane Couette flow, J. Fluid Mech. 229, 499–516
Nagata, M. (1990): Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity, J. Fluid Mech. 217, 519–527
Risken, H. (1984): The Fokker-Planck Equation. Methods of Solution and Applications, Springer
Saltzmann, B. (1961): Finite amplitude free convection as an initial value problem I, J. Atmos. Sci. 19, 329
Schmiegel, A. and Eckhardt, B. (1997): Fractal stability border in plane Couette flow, preprint
Tillmark, N. and Alfredsson, P.H. (1992): Experiments on transition in plane Couette flow, J. Fluid. Mech 235, 89–102
Trefethen, L.N., Trefethen, A., Reddy, S. and Driscoll, T. (1993): Hydrodynamic stability without eigenvalues, Science 261, 578
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Eckhardt, B., Marzinzik, K., Schmiegel, A. (1998). Transition to turbulence in shear flows. In: Parisi, J., Müller, S.C., Zimmermann, W. (eds) A Perspective Look at Nonlinear Media. Lecture Notes in Physics, vol 503. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0104973
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DOI: https://doi.org/10.1007/BFb0104973
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