Improving the Theoretical Prediction of Turbulent Quantities in a Turbulent Straight Pipe Flow

The theory of turbulent plane Poiseuille flows, which is at the basis of turbulent flows in straight pipes, has recently been improved by Guo (2016), who proposed in IAHR’s Journal of Hydraulic Research a novel, quartic, analytic profile for the distribution of the eddy viscosity. He deduced an improved analytic velocity profile that deviates from the traditional, simple, log-law and fits better with experimental data in the whole pipe section.

Here, we first notice that Guo’s model allows writing an analytical profile for the production P of TKE (Turbulent Kinetic Energy) that fits DNS (Direct Numerical Simulations) quite well. In order to deduce analytical profiles for the TKE and its dissipation rate epsilon, we make an additional assumption inspired by DNS: a parabolic distribution of the turbulent turnover time scale tau is valid. By using the dimensional relations between tau and TKE and epsilon, we get theoretical profiles for the latter two quantities that fit well the DNS. As an illustration, the distribution of epsilon and P/epsilon as a function of the elevation above the lower wall, non-dimensionalised using the ‘pipe’ half-width, agree very well the data. The present model improves significantly the traditional laws deduced from Prandtl’s mixing length assumption.