Abstract
The Langevin and diffusion equations for statistical velocity and displacement of marked fluid particles are formulated for turbulent flow at large Reynolds number for which Lagrangian Kolmogorov K-41 theory holds. The damping and diffusion terms in these equations are specified by the first two terms of a general expansion in powers of \(C_{0}^{-1}\) where C0 is Lagrangian based universal Kolmogorov constant: \(6\lesssim C_{0}\lesssim 7\). The equations enable the derivation of descriptions for transport by turbulent fluctuations of conserved scalars, momentum, kinetic energy, pressure and energy dissipation as a function of the derivative of their mean values. Except for pressure and kinetic energy, the diffusion coefficients of these relations are specified in closed-form with \(C_{0}^{-1}\) as constant of proportionality. The relations are verified with DNS results of channel flow at Reτ=2000. The presented results can serve to improve or replace the diffusion models of current CFD models.
Highlights
A general statistical description of turbulent flow has yet to be delivered [1, 2]
Studies in the past decade confirm several of the universal statistical properties of Lagrangian Kolmogorov Theory [6,7,8,9]
The focus is on a statistical description of the large scale variables which starts from the Markov model and which is applicable to configurations of turbulent flow which are otherwise as general as possible
Summary
A general statistical description of turbulent flow has yet to be delivered [1, 2]. What is known are partial results: see e.g. [1, 3]. An old and still widely used approach is to average the Navier-Stokes equations and to introduce a semi-empirical description for turbulent momentum and scalar transport It involves implementing some form of gradient hypothesis as answer to the closure problem. The focus is on a statistical description of the large scale variables which starts from the Markov model and which is applicable to configurations of turbulent flow which are otherwise as general as possible. The diffusion equation for the position of a marked fluid particle entails an Eulerianbased description of turbulent transport of a conserved scalar It was derived from the Langevin equation for statistical fluid particle velocity which is a Lagrangian-based description.
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