Langevin and diffusion equation of turbulent fluid flow

A derivation of the Langevin and diffusion equations describing the statistics of fluid particledisplacement and passive admixture in turbulent flow is presented. Use is made of perturbationexpansions. The small parameter is the inverse of the Kolmogorov constant C0, which arises fromLagrangian similarity theory. The value of C0 in high Reynolds number turbulence is 5–6. Toachieve sufficient accuracy, formulations are not limited to terms of leading order in C0−1 includingterms next to leading order in C0−1 as well. Results of turbulence theory and statistical mechanics areinvoked to arrive at the descriptions of the Langevin and diffusion equations, which are unique upto truncated terms of O??C0−2?? in displacement statistics. Errors due to truncation are indicated toamount to a few percent. The coefficients of the presented Langevin and diffusion equations arespecified by fixed-point averages of the Eulerian velocity field. The equations apply to generalturbulent flow in which fixed-point Eulerian velocity statistics are non-Gaussian to a degree ofO??C0−1??. The equations provide the means to calculate and analyze turbulent dispersion of passive oralmost passive admixture such as fumes, smoke, and aerosols in areas ranging from atmosphericfluid motion to flows in engineering devices.