The Problem of Turbulence and the Manifold of Asymptotic Solutions of the Navier-Stokes Equations
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After a brief introduction to the concepts of energy stability, global stability and linear stability three theoretical approaches to different cases of hydrodynamic turbulence under stationary external conditions are discussed. Phase turbulence may occur in the weakly nonlinear limit of the Navier-Stokes-equations of motion when the instability of the basic primary state occurs in the form of a highly degenerate bifurcation as, for example, in the case of convection in a fluid layer heated from below and rotating about a vertical axis. Another quite generally applicable approach towards understanding turbulent fluid flow and its coherent structures in particular is the sequence-of-bifurcations approach. Discrete transitions from simple to complex fluid flows can be analyzed in the form of successive bifurcations when the system exhibits a maximum of symmetries based on the assumption of homogeneity in two spatial dimensions. The spatially periodic solutions generated through the sequence-of-bifurcations approach may not be realized in experimental situations where inhomogeneous onsets of disturbances may lead to chaotic fluid flows long before all spatially periodic solutions become unstable. But the tertiary and quaternary solutions exhibit in the clearest way the dynamic mechanisms that operate in turbulent states of flow. Finally the theory of bounds for turbulent transports in the asymptotic range of high Reynolds and Rayleigh numbers is outlined.
KeywordsRayleigh Number Asymptotic Solution Couette Flow Fluid Layer Steady Solution
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