Loop Equation in Turbulence

Details Loop Equation in Turbulence Loop Equation in Turbulence

1993-03-23

arxiv.org/abs/hep-th/9303130v2

Physics

High Energy Physics

High Energy Physics - Theory

33 pages, 7 figures, LaTeX

Scientific paper

The incompressible fluid dynamics is reformulated as dynamics of closed loops $C$ in coordinate space. This formulation allows to derive explicit functional equation for the generating functional $\Psi[C]$ in inertial range of spatial scales, which allows the scaling solutions. The requirement of finite energy dissipation rate leads then to the Kolmogorov index. We find an exact steady solution of the loop equation in inertial range of the loop sizes. The generating functional decreases as $\EXP{-A^{\tt}}$ where $A=\oint_C r \wedge dr$ is the area inside the loop. The pdf for the velocity circulation $\Gamma$ is Lorentzian, with the width $\bar{\Gamma} \propto A^{\tt} $.

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