Abstract
A comprehensive summary and update is given of Brouwers’ statistical model that was developed during the previous decade. The presented recapitulated model is valid for general inhomogeneous anisotropic velocity statistics that are typical of turbulence. It succeeds and improves the semiempirical and heuristic models developed during the previous century. The model is based on a Langevin and diffusion equation of which the derivation involves (i) the application of general principles of physics and stochastic theory; (ii) the application of the theory of turbulence at large Reynolds numbers, including the Lagrangian versions of the Kolmogorov limits; and (iii) the systematic expansion in powers of the inverse of the universal Lagrangian Kolmogorov constant C0, C0 about 6. The model is unique in the collected Langevin and diffusion models of physics and chemistry. Presented results include generally applicable expressions for turbulent diffusion coefficients that can be directly implemented in numerical codes of computational fluid mechanics used in environmental and industrial engineering praxis. This facilitates the more accurate and reliable prediction of the distribution of the mean concentration of passive or almost passive admixture such as smoke, aerosols, bacteria, and viruses in turbulent flow, which are all issues of great societal interest.
Highlights
IntroductionModels of turbulent flow were of a rather elementary form. Focus was on the contribution of fluctuations on mean flow quantities
Model of Turbulent DispersionEarly models of turbulent flow were of a rather elementary form
A healthy starting point is the formulation of a Langevin equation for particle velocity, e.g., van Kampen [16]. As this approach serves as a guidance to the formulation of a Langevin model for turbulent dispersion, we summarize the steps in the solution of the molecular problem
Summary
Models of turbulent flow were of a rather elementary form. Focus was on the contribution of fluctuations on mean flow quantities. More advanced models concerned statistical descriptions of the fluctuations themselves They were restricted to isotropic homogeneous turbulence and were used to predict dispersion, e.g., Taylor [4] and Batchelor [5]. The present paper gives a comprehensive summary and update of this work It positions the model in the broader context of statistical modelling and the physics of random molecular motion. Focus is on the main results, their fundamental basis, and the general nature for describing dispersion by large scales of turbulence. Their potential for application in and improvement of codes of computational fluid dynamics used in environmental and industrial engineering is demonstrated.
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