ISSN 1028-3358, Doklady Physics, 2012, Vol. 57, No. 8, pp. 312–316. © Pleiades Publishing, Ltd., 2012. Original Russian Text © I.I. Vigdorovich, 2012, published in Doklady Akademii Nauk, 2012, Vol. 445, No. 5, pp. 516–520. 312 The self-similar flow in a turbulent boundary layer, which is in a state close to separation as a result of the effect of adverse pressure gradient, was investigated. Such a boundary layer has a triple-deck asymptotic structure. Between outer and near-wall regions above the logarithmic sublayer, i.e., the constant-stress layer, an intermediate region—the gradient sublayer—is formed, the shear stress in which varies linearly due to adverse longitudinal pressure gradient. In the external part of the gradient sublayer, the velocity profile obeys the square-root law. The velocity profile obtained from the solution for the outer region satisfies a slip condi- tion on the wall. The slip value decreases as the simi- larity parameter Ω = increases and vanishes at the value of Ω = 0.0911 to which the separation cor- responds, here δ ∗ is the displacement thickness, and U and U ' are the free-stream velocity and its derivative with respect to the longitudinal coordinate. In this case, the exponent m in the law specifying the self-similar velocity distribution in the external flow increases, with separation occurring not at the minimal value of m = – , which corresponds to the strongest adverse pressure gradient, but at the value m = –0.228. 1. We consider the flow of an incompressible fluid in the turbulent boundary layer on a flat plate for the power dependence of the free-stream velocity on the longitudinal coordinate: (1) where x is the distance from the leading edge of the plate, and B is a certain dimensional constant. In [1] it δ * U' U --------- – 1 3 - Ux () Bx m , x 0 , ≥ = is shown that the turbulent shear stress in the boundary layer can be presented in this case as (2) Here, y is the distance to the wall, u is the longitudinal component of the mean velocity, Δ is the boundary- layer thickness, and S is a continuous function of three variables satisfying the condition S(∞, 0, m) = κ 2 , where κ is the von Karman constant. Relation (2) is a consequence of the fact that the turbulent flow in the self-similar boundary layer depends only on three gov- erning parameters: the constants B and m included in Eq. (1) and the kinematic viscosity ν of the fluid. Similar relations are valid for other components of the Reynolds stress tensor: (3) The stream function ψ(x, y) of the mean flow satis- fies the equation which is more exact than the normal boundary-layer equation because it takes into account the change of pressure across the layer in the first approximation. This equation in new variables [2] with taking into account closure conditions (2) and (3) becomes u ' v ' 〈 〉 y ∂ u ∂ y ---- ⎝ ⎠ ⎛ ⎞ 2 S Re η m , , ( ) , – = Re y 2 ν --- ∂ u ∂ y ---- , η y Δ -- . = = u ' 2 〈 〉 y ∂ u ∂ y ---- ⎝ ⎠ ⎛ ⎞ 2 S 1 Re η m , , ( ) , = v ' 2 〈 〉 y ∂ u ∂ y ---- ⎝ ⎠ ⎛ ⎞ 2 S 2 Re η m , , ( ) . = ψ y ψ xy ψ x ψ yy – UU' νψ yy u ' v ' 〈 〉 – ( ) y + = + v ' 2 〈 〉 u ' 2 〈 〉 – ( ) x , ψ UΔΨ ξ η , ( ) , ξ Re Δ , Re Δ ln UΔ ν ------ = = = MECHANICS A Self-Similar Turbulent Boundary Layer in a State Close to Separation I. I. Vigdorovich Presented by Academician V.A. Levin April 9, 2012 Received April 12, 2012 DOI: 10.1134/S1028335812080083 Institute of Mechanics, Moscow State University, Moscow, 117192 Russia e-mail: vigdorovich@imec.msu.ru