Received: 3 November 2016 Revised: 1 October 2017 Accepted: 8 December 2017 DOI: 10.1002/zamm.201600246 ORIGINAL PAPER Fast and accurate modelling of frictional transient pipe flow Kamil Urbanowicz West Pomeranian University of Technology, Al Piastów 17, Szczecin, Poland Correspondence K. Urbanowicz, West Pomeranian Univer- sity of Technology, Al Piastów 17, Szczecin, Poland. Email: kurbanowicz@zut.edu.pl This paper is devoted to the one-dimensional (1D) modelling of hydraulic losses dur- ing transient flow of liquids in pressure lines. Unsteady pipe wall shear stress is present in the form of a convolution of liquid acceleration with a weighting function. The weighting function depends on the dimensionless time and the Reynolds number. In the original model of Zielke (1968), computation of the convolution integral had a complex and inefficient mathematical structure (featured power growth of computa- tional time). Therefore, further work aimed at developing efficient models for esti- mation of unsteady hydraulic resistance (with nearly linear growth of computational time). In the present paper, a correction to the erratic recursive formula by Schohl (1993), being used to calculate the unsteady wall shear stresses during transient flow, is presented. The simulation results obtained with the corrected Schohl's recursive for- mula are consistent with the classic but computationally inefficient formula proposed recently by Vardy and Brown (2010). The accuracy of the efficient weighting func- tion obtained in wall shear stress models is verified. The results of pressure pulsation obtained when taking into account cavitating flow, or not, are surprising in the sense that the weighting function does not need to be built by a lengthy sum of exponential terms to accurately simulate the transient event. KEYWORDS cavitation, convolution integral, hydraulic resistance, transient pipe flow, weighting function 1 INTRODUCTION The modelling of transient liquid flow in pipes (water hammer, accelerated and decelerated flows, etc.) is not a simple task. One might need to take into account the concomitants: fluid-structure interaction (FSI), [1–6] cavitation, [7–29] frequency-dependent hydraulic resistance, [30–62] and viscoelasticity of pipes. [63–68] In this paper, the flow in restrained rigid metal pipes will be analysed; then, FSI and viscoelastic properties of pipe may be neglected. The classic Streeter's [7] discrete vapour cavity model (DVCM) allows vapour cavities to form at all computational sections when the pressure falls to the liquid vapour pressure. The main assumptions in this model are as follows: pressure in the cavity is set to the constant liquid vapour pressure, maximum volume of cavity should be considered much smaller than the volume of liquid in the reach of pipe, pressure waves are reflected off of the cavity when it exists, cavity does not moves and contain pure liquid vapour, the isothermal condition prevail in cavities and the mass and momentum of the formed liquid vapour are negligible. As the classic DVCM generates unrealistic pressure pulses called “spikes”, [11] several authors attempted to suppress them. Kranenburg [15] used a numerical filter to suppress spurious oscillations. Safwat-Polder [32] avoided problems with multi- cavity collapse allowing the cavity zone to form only at one predetermined location of pipe. Kot-Youngdahl, [16] Miwa et al. [17] and later Anderson-Araie [18] introduced a additional numerical damping (as the effect of interpolation) to minimise the effect of “spikes”. Streeter [19] proposed the so called “consolidation” method which significantly reduced the pressure spikes, but it was too complex to be incorporated into a general programme. This method was used and corrected first by Simpson, [20] and later Z Angew Math Mech. 2018;1–22. © 2018 WILEY-VCH Verlag GmbH &Co. KGaA, Weinheim 1 www.zamm-journal.org