A model of a turbulent boundary layer with a nonzero pressure gradient

According to a model of the turbulent boundary layer that we propose, in the absence of external turbulence the intermediate region between the viscous sublayer and the external flow consists of two sharply separated self-similar structures. The velocity distribution in these structures is described by two different scaling laws. The mean velocity u in the region adjacent to the viscous sublayer is described by the previously obtained Reynolds-number-dependent scaling law φ = u / u _* = A η ^α , A = 1/ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{equation*}\sqrt{3}\end{equation*} ln Re _Λ + 5/2, α = 3/2 In Re _Λ , η = u _* y /ν. (Here u _* is the dynamic or friction velocity, y is the distance from the wall, ν the kinematic viscosity of the fluid, and the Reynolds number Re _Λ is well defined by the data.) In the region adjacent to the external flow, the scaling law is different: φ = B η ^β . The power β for zero-pressure-gradient boundary layers was found by processing various experimental data and is close (with some scatter) to 0.2. We show here that for nonzero-pressure-gradient boundary layers, the power β is larger than 0.2 in the case of an adverse pressure gradient and less than 0.2 for a favorable pressure gradient. Similarity analysis suggests that both the coefficient B and the power β depend on Re _Λ and on a new dimensionless parameter P proportional to the pressure gradient. Recent experimental data of Perry, Marušić, and Jones were analyzed, and the results are in agreement with the model we propose.