A common method to calculate the flow rate and consequently hydraulic efficiency in hydropower plants is the pressure-time method. In the present work, the pressure-time method is studied numerically by three-dimensional (3D) simulations and considering the change in the pipe cross section (a contraction). Four different contraction angles are selected for the investigations. The unsteady Reynolds-averaged Navier–Stokes (URANS) equations and the low-Reynolds k–ω shear stress transport (SST) turbulence model are used to simulate the turbulent flow. The flow physics in the presence of the contraction, and during the deceleration period, is studied. The flow rate is calculated considering all the losses: wall shear stress, normal stresses, and also flux of momentum in the flow. The importance of each term is evaluated showing that the flux of momentum plays a most important role in the flow rate estimation while the viscous losses term is the second important factor. To extend the viscous losses calculations applicability to real systems, the quasi-steady friction approach is employed. The results showed that considering all the losses, the increase in the contraction angle does not influence the calculated errors significantly. However, the use of the quasi-steady friction factor introduces a larger error, and the results are reliable approximately up to a contraction angle of ϴ = 10 deg. The reason imparts to the formation of a local recirculation zone upstream and inside the contraction, which appears earlier as the contraction angle increases. This feature cannot be captured by the quasi-steady friction models, which are derived based on the fully developed flow assumption.
Concerning the shear stress for bent elastic rod with variable cross-section
