Analytical expressions for effective weighting functions used during simulations of water hammer

Abstract

For some time, work has been underway aimed at significant simplification of the modelling of hydraulic resistance occurring in the water hammer while maintaining an acceptable error. This type of resistance is modelled using a convolution integral, among others, from local acceleration of a liquid and a certain weighting function. The recently completed work shows that during efficient calculations of the convolution integral, the effective weighting function used does not have to be characterised by large convergence with a classical function (according to Zielke during laminar flow and to Vardy-Brown during turbulent flow). However, it must be a sum of at least two or three exponential expressions so that the final results of the simulation could be considered as satisfactory. In this work, it has been decided to present certain analytical formulas using which it will be possible to determine the coefficients of simplified effective weighting functions in a simple direct way.

Water Hammer Investigation of the Shut-Down of a High-Head Hydropower Plant at Very High Reynolds Number Flows

“…Dimensionless times used in CBM (see Section 1.1) for the tunnel, and Penstocks I, II, and III are Δτ = {0.0069×10 -6 , 0.0414×10 -6 , 0.0342×10 -6 , 0.0242×10 -6 }, respectively. Consequently, the Urbanowicz approximation model in CBM [21] and [22] has been used. Duan's parameter [14] for the tunnel and three penstocks is I D = {0.014, 0.0834, 0.0847, 0.03}.…”

“…The widely used approximation of Vítkovský et al [20] is accurate over a broad range of dimensionless times Δτ = Dt4ν/D 2 [10 -6 , 10 -1 ]. For lower Δτ values, Urbanowicz [21] and [22] developed a computationally efficient and accurate approximation of weighting functions that should be used when Δτ ≤ 10 -6 .…”

Transient Liquid Flow in Plastic Pipes

“…The m i and n i coefficients are functions of the dimensionless time step Δt  . Their values can be determined with use of the analytical formulas presented in [35]. The range of applicability of the filtered weighting function is from Δt  up to 10 3 Δt  .…”

Vibration transients of reservoir-pipe-valve system caused by water hammer

“…Zarzycki and Urbanowicz reported a series of important works about the unsteady pipe flow. They proposed methods to confirm coefficients of weighting functions in a simple way (Urbanowicz, 2017), approximate weighting functions as sums of exponential components and improved the lumping friction model estimating the basic parameters (Urbanowicz and Zarzycki, 2015). They also simulated transients of a turbulent pipe flow (Zarzycki et al, 2007) and a cavitating liquid flow (Urbanowicz et al, 2012; Urbanowicz and Zarzycki, 2008;Zarzycki and Urbanowicz, 2008).…”