NONLOCAL STOCHASTIC MIXING-LENGTH THEORY AND THE VELOCITY PROFILE IN THE TURBULENT BOUNDARY-LAYER

Turbulence mixing by finite size eddies will be treated by means of a novel formulation of nonlocal K-theory, involving sample paths and a s tochastic 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 re lation with other nonlocal turbulence models (e.g, transilience and sp ectral diffusivity theory) is also discussed. Using an analytical samp ling rate model (satisfying exchange) the theory is applied to the bou ndary layer (using a scaling hypothesis), which maps boundary layer tu rbulence mixing of scalar densities onto a nondiffusive (Kubo-Anderson or kangaroo) type stochastic process. The resulting transport equatio n for longitudinal momentum P-x = rho ($) over bar U is solved for a u nified description of both the inertial and the viscous sublayer inclu ding the crossover. With a scaling exponent epsilon approximate to 0.5 8 (while local turbulence would amount to epsilon --> infinity) the ve locity profile ($) over bar(+) = f(y(+)) is found to be in excellent a greement with the experimental data. Inter alia (i) the significance o f epsilon as a turbulence Canter set dimension, (ii) the value of the integration constant in the logarithmic region (i.e, if y(+) --> infin ity), (iii) linear timescaling, and (iv) finite Reynolds number effect s will be investigated. The (analytical) predictions of the theory for near-wall behaviour (i.e, if y(+) --> 0) of fluctuating quantities al so perfectly agree with recent direct numerical simulations.