4.4.2 The Chilton-Colburn analogy
The Reynolds analogy does not always give satisfactory results. Thus, Chilton and Colburn experimentally modified the Reynolds’ analogy. The empirically modified Reynolds’ analogy is known as Chilton-Colburn analogy and is given by eq.4.35,

It can be noted that for unit Prandtl number the Chilton-Colburn analogy becomes Reynolds analogy.

4.4.3 The Pradntl analogy
In the turbulent core the transport is mainly by eddies and near the wall, that is laminar sub-layer, the transport is by molecular diffusion. Therefore, Prandtl modified the above two analogies using universal velocity profile while driving the analogy (eq. 4.36).

4.4.4 The Van Karman analogy

Though Prandtl considered the laminar and turbulent laminar sublayers but did not consider the buffer zone. Thus, Van Karman included the buffer zone into the Prandtl analogy to further improve the analogy.