Self-Diffusion by Multivariate-Normal Turbulent Velocity Field

A closed set of exact equations describing statistical theory of turbulent self-diffusion by multivariate-normal turbulent velocity field is derived. In doing so, we first suggest exact formulas for correlations 〈fi(p)fj(p ′)R[f ]〉, 〈g(p)R[f ]〉 and 〈g(p)fj(p ′)R[f ]〉 when the functional R[f ] is functional of functions fi’s having multivariate-normal distribution, g and fi’s have joint normal distribution and zero mean values. Originative works of Taylor [1] and Batchelor [2] on turbulent self-diffusion of fluid particles in Lagrangian and Eulerian frameworks, respectively, have placed statistics of Lagrangian velocity field, first put forward by Taylor [1], as fundamental properties in the field of statistical theory of diffusion. Despite many persistent efforts (for review, see e.g., Refs. [3, 4]), the theory of diffusion remains incomplete mainly due to involved closure problems [4, 5, 6] and lack of accurate prediction of Lagrangian statistics [7, 8, 9] in general turbulent flows. In this letter, we base our Eulerian analysis on the equation for a passive scalar field ψ(x, t) whose ensemble average (denoted by 〈 〉) denotes Green’s function G for the evolution in space-time (x − t) of an arbitrary initial probability distribution for the position of marked fluid particles in space [2, 5, 8]. We solve the closure problem arising in the equation for 〈ψ(x, t)〉 resulting in an appearance of Lagrangian velocity correlations. We then obtain equations relating Lagrangian and Eulerian correlations for the velocity field. These equations are exact when the fluctuations u′i in turbulent velocity field ui(x, t) = Ui(x, t) + u ′ i(x, t) over the mean velocity Ui(x, t) obey multivariatenormal distribution. In doing so, we first suggest three new functional formulas which along with the Furutsu-Donsker-Novikov functional formula [10] are used in our Eulerian analysis. These four functional formulas are now discussed briefly before presenting the Eulerian analysis. Email address: [email protected]