Spherical model for turbulence

Citation

Mou, Chung-Yu (1993) Spherical model for turbulence. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/SQXS-2H17. https://resolver.caltech.edu/CaltechTHESIS:12262012-133726544

Abstract

A new set of models for homogeneous, isotropic turbulence is considered in which the Navier-Stokes equations for incompressible fluid flow are generalized to a set of N coupled equations in N velocity fields. It is argued that in order to be useful these models must embody a new group of symmetries, and a general formalism is laid out for their construction. The work is motivated by similar techniques that have had extraordinary success in improving the theoretical understanding of equilibrium phase transitions in condensed matter systems. The key result is that these models simplify when N is large. The so-called spherical limit, N → ∞ can be solved exactly, yielding a closed pair of nonlinear integral equations for the response and correlation functions. These equations, known as Kraichnan's Direct Interaction Approximation (DIA) equations, are, for the first time, solved fully in the scale-invariant turbulent regime, and the implications of these solutions for real turbulence (N = 1) are discussed. In particular, it is argued that previously applied renormalization group techniques, based on an expansion in the exponent, y, that characterizes the driving spectrum, are incorrect, and that the Kolmogorov exponent ς has a nontrivial dependence on N, with ς(N → ∞) = 3/2 This value is remarkably close to the experimental result, ς ≈ 5/3, which must therefore result from higher order corrections in powers of 1/N. Prospects for calculating these corrections are briefly discussed: though daunting, such a calculations would, for the first time, provide a controlled perturbation expansion for the Kolmogorov, and other, exponents. Our techniques may also be applied to other nonequilibrium dynamical problems, such as the KPZ equation for interface growth, and perhaps to turbulence in nonlinear wave systems.

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