ISSN 0015-4628, Fluid Dynamics, 2017, Vol. 52, No. 5, pp. 631–645.  Pleiades Publishing, Ltd., 2017. Original Russian Text  I.I. Vigdorovich, 2017, published in Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, 2017, No. 5, pp. 38–52. Turbulent Thermal Boundary Layer on a Plate. Reynolds Analogy and Heat Transfer Law over the Entire Range of Prandtl Numbers I. I. Vigdorovich Institute of Mechanics, Lomonosov Moscow State University, Michurinskii pr. 1, Moscow, 119192 Russia e-mail: vigdorovich@imec.msu.ru Received February 17, 2017 Abstract—A rational asymptotic theory is proposed, which describes the turbulent dynamic and thermal boundary layer on a flat plate under zero pressure gradient. The fact that the flow depends on a finite number of governing parameters makes it possible to formulate algebraic closure conditions relating the turbulent shear stress and heat flux with the gradients of the averaged velocity and temperature. As a result of constructing an exact asymptotic solution of the boundary layer equations, the known laws of the wall for velocity and temperature, the velocity and temperature defect laws, and the expressions for the skin friction coefficient, Stanton number, and Reynolds analogy factor are obtained. The latter makes it possible to give two new formulations of the temperature defect law, one of which is identical to the velocity defect law and contains neither the Stanton number nor the turbulent Prandtl number, and the second formulation does not contain the skin friction coefficient. The heat transfer law is first obtained in the form of a universal functional relationship between three parameters: the Stanton number, the Reynolds number, and the molecular Prandtl number. The conclusions of the theory agree well with the known experimental data. Keywords: turbulent boundary layer, similarity laws, Reynolds analogy, temperature defect law, friction and heat transfer laws. DOI: 10.1134/S0015462817050052 The well-known temperature defect law, valid in the outer region of turbulent boundary layer and in the central part of a circular tube or a flat channel (see, for instance, [1]), uses the so-called friction temperature as a characteristic scale. This friction temperature is calculated in terms of the skin friction coefficient and the Stanton number, which requires measuring the shear stress and thermal flux on the wall. A similar statement is true for the universal heat transfer law [1], which for a fixed value of the molecular Prandtl number relates three parameters, namely, the Stanton number, the skin friction coefficient, and the Reynolds number. In a pipe or channel flow, it is easy to calculate the shear stress on the wall based on measuring the longitudinal pressure drop, but in the boundary layer on a plate (considered in this paper), to the best of author’s knowledge, simultaneous measurements of the friction and heat flux were performed only in [2–4]. In deriving the above-mentioned similarity laws [1], the authors used only the dimension analysis and the only physical assumption [5], according to which the considered flow has two characteristic length scales: viscous one (which determines the thickness of a viscous sublayer on the wall) and outer one (boundary layer thickness or pipe radius). For large Reynolds numbers, the molecular viscosity and thermal conductivity outside the viscous sublayer are insignificant (they do not enter in the governing parameters of the flow outside the viscous sublayer), and the outer scale has no impact on the processes near the wall and is not a governing parameter in this region. In this paper, we propose a different approach to this classic problem. Our approach is based on solving the equations of motion and heat transfer, in which, under the same physical assumption [5], the closure con- 631