Recovery factor turbulent flow

For turbulent flow, with roughly uniform distribution, assuming a constant fricdion factor, the combined effect of friction and inerrtal (momentum) pressure recovery is given by... 

In order to find the Reynolds number, the surface temperature must be known. However, in order to find the wall temperature, the recovery factor must be known and its value is different in laminar and turbulent flow. Therefore, an assumption as to the nature of the flow, i.e., laminar or turbulent, will be made. The wall temperature and then the gas properties will be found and then the Reynolds number, i.e. ... 

We must consider the laminar and turbulent portions of the boundary layer separately because the recovery factors, and hence the adiabatic wall temperatures, used to establish the heat flow will be different for each flow regime. It turns out that the difference is rather small in this problem, but we shall follow a procedure which would be used if the difference were appreciable, so that the general method of solution may be indicated. The free-stream acoustic velocity is calculated from... 

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