Fractals and Multi-fractals in Turbulence
Turbulent flows refer to situations in which the flow properties at any point vary in a statistically random manner. Fourier analysis shows that wave fluctuations in a range of frequencies and wave numbers are present, the width of the range changing with certain flow parameters like the Reynolds number. (Attempts at characterizing turbulence structure are hampered by the complexity associated with strong nonlinear couplings over a wide range of scales. This situation is somewhat made simpler at high Reynolds numbers due to separation of scales in this limit.) The various component motions interact through the nonlinear terms in the equations of motion, and the observed properties of the turbulence are thought of as being the statistical result of such interactions.
Mathematically, the description of these interactions should center on invariant measures. Although, there is no rigorous theory about the existence of strictly invariant measures in turbulence, experimental observations strongly support the idea that turbulence at small scales organizes itself into a statistically stationary universal state. Apparently, as the Reynolds number of the flow becomes infinite, all the invariance properties of the Navier-Stokes equations, possibly broken by the mechanisms producing the turbulence, are recovered asymptotically at small scales in a statistical sense.
KeywordsEnergy Dissipation Hausdorff Dimension Energy Dissipation Rate Inertial Range Singularity Spectrum
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