Scaling laws for turbulent boundary layer flows in favorable and adverse pressure gradient

Abstract

An analytical theory is proposed to describe incompressible plane and axisymmetric turbulent boundary layer flows in favorable and adverse pressure gradients for near-equilibrium conditions. Scaling laws for the mean velocity, the Reynolds stress components, and the skin friction have been established. A universal friction law makes it possible to represent the skin friction distributions corresponding to different Reynolds numbers and pressure gradients in terms of a function of one variable. The theory is based on general physical assumptions and does not involve any special hypotheses on the nature of the turbulent motion. Formulation of the problemWe consider a two-dimensional incompressible turbulent boundary layer flow on a plane profile or an axisymmetric body. In the latter case, we consider a thick body when the boundary layer thickness is much less than the distance from the axis of symmetry. We show that in the general case, the shear stress can be represented in the formHere, y is the distance from the wall, u is the rotational component of the longitudinal mean velocity in the boundary layer, i. e., the difference between the full mean velocity and its irrotational part, ∆ is the boundary layer thickness, U e is the velocity at the outer edge of the boundary layer, r is the radius specifying the body shape in the axisymmetric case (j = 1), and T is a usual function of the four variables and a functional of a curve L in the phase plane (z 1 , z 2 ) defined parametrically on the basis of the function γ(ξ) as z 1 = dγ/dξ, z 2 = d 2 γ/dξ 2 . Closure condition (1) is derived without invoking any special hypotheses on the nature of the turbulent motion only from the reasoning of dimensional invariance [1]. Representations analogous to (1) are also valid for the other Reynolds stress components.The Reynolds-averaged Navier-Stokes equations along with the closure conditions give us a well-posed boundary-value problem for the mean velocity field. We rewrite the equations for a dimensionless stream function Ψ(ξ, η) and seek the solution as ξ → ∞ [2,3]. Thus, the small parameter in the problem is the reciprocal of the logarithm of the Reynolds number based on a characteristic boundary layer thickness. In this limit process, the derivatives with respect to ξ can be neglected and in the outer region of the boundary layer, the problem is reduced to an ordinary differential equation for the function f (η) related to the velocity defectHere, L 0 is the point in the phase plane with coordinates (0, 0). In this limit process, the functional T specifying in equation (2) the turbulent shear stress degenerates to a usual function of two variables. Equation (2) depends on a single parameter γ, which varies within finite limits: −1 < γ ≤ 1. The solution has a singularity as γ → −1. In this case, the outer region of the boundary layer takes a two-layer structure. An intermediate region where the velocity obeys the square root law appears above the logarithmic sublayer which is typical of the flows at strong adverse pressu...