Velocity Distribution in Rough Pipe: the Model Based on the Analytical Description of Resistance Curves in Nikuradse’s Experiments

Calculation and optimization of such complex systems as ventilating networks, water supply systems, etc. taking into account roughness of a surface and possible processes of heat and mass transfer demands use simple hydrodynamic models, which are at the same time sufficiently exact for engineering calculations. The goal of this paper is to develop such model. Prandtl’s model of a two-layer flow in the smooth pipe with the linear law of velocity distribution for a near-wall laminar layer and the logarithmic law for a turbulent core is used as a basis, which spreads to a case of rough pipes. The velocity profile equation and also the equations for the coordinate Y b of the laminar and turbulent border and its velocity U b are derived. This paper is based on the use of the analytical description of Nikuradse’s experimental curves λ (Re, δ ) ( λ is the resistance coefficient, δ is the relative roughness) which earlier was found by the author. The dependence λ (Re*, δ ) (Re* is the dynamic Reynolds number) is included into all above-mentioned equations. Results of numerical calculations of Y b and U b as functions of Re* for the values of δ which are the same as in Nikuradse’s experiments are given. The calculation of velocity profiles for some values of Re * and δ is also fulfilled. Comparisons with the experimental velocity profiles obtained by Nikuradse demonstrate a good agreement.