Further Developments in Rapidly Decelerating Turbulent Pipe Flow Modeling

In the last two decades, energy dissipation in unsteady-state pressurized pipe flow has been examined by various authors, where the instantaneous wall shear stress is split into a quasi-steady and an unsteady shear stress component. The focus of most past studies is on formulating expressions for the unsteady wall shear stress, but there has been less work on the key parameters governing the dominance of unsteady friction in transient flows. This paper derives an expression for the head envelope damping for turbulent flows in smooth and rough pipes and provides new and carefully measured field data for the initial (i.e. pre-transient) Reynolds number, Re0, that ranges from 97000 to 380000. The analytical solutions is derived on the basis of one-dimensional (1-D) waterhammer equations in which the unsteady component is represented by existing convolutional unsteady friction formulas for both smooth and rough turbulent sub-regimes. The analytical solution is used to formulate general, encompassing and theoretically-based dimensionless parameters to assess the importance of unsteady friction in comparison to the quasi-steady component. In addition, the analytical solution furnishes the similitude relations that allowed the damping behavior from existing laboratory tests, the field tests conducted as part of this research and the weighting function-based (WFB) models to be investigated and compared in a coherent manner in a single graph. The analysis confirms that the magnitude of Re0 has a significant impact on the damping for transients generated by flow stoppage. In addition, the results show that convolutional unsteady friction model that uses the frozen eddy viscosity hypothesis and Re0 has accuracy that decreases with time. An improvement for this shortcoming is proposed and verified and involves the use of the instantaneous Reynolds number in lieu of the pre-transient Reynolds number in the evaluation of the WFB models. The result is a modified unsteady friction model that provides improved matches for both laboratory and field data compared with the original model.