Nonlocal stochastic mixing-length theory and the velocity profile in the turbulent boundary layer

Abstract

Turbulence mixing by finite size eddies will be treated by means of a novel formulation of nonlocal K-theory, involving sample paths and a stochastic closure hypothesis, which implies a well defined recipe for the calculation of sampling and transition rates. The connection with the general theory of stochastic processes will be established. The relation with other nonlocal turbulence models (e.g. transilience and spectral diffusivity theory) is also discussed. Using an analytical sampling rate model (satisfying exchange) the theory is applied to the boundary layer (using a scaling hypothesis), which maps boundary layer turbulence mixing of scalar densities onto a nondiffusive (Kubo-Anderson or kangaroo) type stochastic process. The resulting transpport equation for longitudinal momentum P x ≡ ϱ U is solved for a unified description of both the inertial and the viscous sublayer including the crossover. With a scaling exponent ε ≈ 0.58 (while local turbulence would amount to ε → ∞) the velocity profile U + = ƒ(y +) is found to be in excellent agreement with the experimental data. Inter alia (i) the significance of ε as a turbulence Cantor set dimension, (ii) the value of the integration constant in the logarithmic region (i.e. if y + → ∞), (iii) linear timescaling, and (iv) finite Reynolds number effects will be investigated. The (analytical) predictions of the theory for near-wall behaviour (i.e. if y + → 0) of fluctuating quantities also perfectly agree with recent direct numerical simulations.