Results of a Variational Principle for Path Integral Functionals Describing Stationary Turbulence
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概要
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A Lagrangian description derived from the path probability density for nonlinear, driven stochastic Markov processes is given for velocity fluctuations in fully developed stationary turbulence governed by the Navier-Stokes equations for incompressible fluids subject to random Gaussian stirring forces. An inequality based on the convexity of the weight functional for different paths is used to apply a variational principle to the generating functional for correlation functions. The physical character of the resulting mean field type approximation is discussed. A wavenumber dependent eddy viscosity simulating the energy transfer effect caused by nonlinear triad interactions between different Fourier velocity modes is determined via a nonlinear integral equation for the energy spectrum E(k) of velocity fluctuations. A cutoff is introduced into the integral kernel which ensures that stirring forces exert strain only on scales larger than the size of the eddy subject to a relaxation process. The influence of long wave length and small scale characteristics of the stirring mechanism on the energy spectrum is investigated. The solution of the integral equation for E(k) obtained numerically for various stirring mechanisms is explained with analytical approximations for small and large wavenumbers. The results are compared with renormalization group calculations for long wavelength spectra and with closure approximations.
- 理論物理学刊行会の論文
- 1979-04-01