Reynolds analogy turbulent boundary layer flow

Numerically determine the local Nusselt number variation with two-dimensional turbulent boundary layer air flow over an isothermal flat plate for a maximum Reynolds number of 107. Assume that transition occurs at a Reynolds number of 5 X 105. Compare the numerical results with those given by the Reynolds analogy. 

The eddy diffusitives for momentum and heat, and Ejj, respectively, are not properties of the fluid but depend on the conditions of flow, especially on all factors that affect turbulence. For simple analogies, it is sometimes assumed that and jf are both constants and equal, but when determined by actual velocity and temperature measurements, both are found to be functions of the Reynolds number, the Prandtl number, and position in the tube cross section. Precise measurement of the eddy diffusivities is diflScult, and not all reported measurements agree. Results are given in standard treatises. The ratio Sh/sm also varies but is more nearly constant than the individual quantities. The ratio is denoted by i/f. For ordinary liquids, where Np > 0.6, is close to 1 at the tube wall and in boundary layers generally and approaches 2 in turbulent wakes. For liquid metals is low near the wall, passes through a maximum of about unity at j/r X 0.2, and decreases toward the center of the pipe. ... 

Early theories for transpiration of air into air [114, 115] were based on the Couette flow approximation. Reference 114 extended the Reynolds analogy to include mass transfer by defining a two-part boundary layer consisting of a laminar sublayer and a fully turbulent core. Here, t = 0 in the sublayer (y < y ), and t = OAy and (i = 0 in the fully turbulent region. The density was permitted to vary with temperature. The effect of foreign gas injection in a low-speed boundary layer was studied in Ref. 116, and all these theories were improved upon in Ref. 117. 

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