Developments in unsteady pipe flow friction modelling

Abstract

This paper reviews a number of unsteady friction models for transient pipe flow. Two distinct unsteady friction models, the Zielke and the Brunone models, are investigated in detail. The Zielke model, originally developed for transient laminar flow, has been selected to verify its effectiveness for "low Reynolds number" transient turbulent flow. The Brunone model combines local inertia and wall friction unsteadiness. This model is verified using the Vardy's analytically deduced shear decay coefficient C* to predict the Brunone's friction coefficient k rather than use the traditional trial and error method for estimating k. The two unsteady friction models have been incorporated into the method of characteristics water hammer algorithm. Numerical results from the quasi-steady friction model and the Zielke and the Brunone unsteady friction models are compared with results of laboratory measurements for water hammer cases with laminar and low Reynolds number turbulent flows. Conclusions about the range of validity for the three friction models are drawn. In addition, the convergence and stability of these models are addressed. RESUMELe papier passe en revue un certain nombre de modeles de friction non permanente en ecoulement transitoire en conduite. Deux modeles de friction non permanente, celui de Zielke et celui de Brunone sont investigues en details. Le modele de Zielke, developpe a l'origine pour les ecoulements transitoires laminaires, a ete selectionne pour tester l'efficacite du modele pour les ecoulements transitoires turbulents a faible nombre de Reynolds. Le modele de Brunone combine la variation de l'inertie locale et de la friction de paroL Ce modele est verifie en utilisant Ie coefficient C* d'amortissement du cisaillement de Vardy dCduit analytiquement pour predire Ie coefficient de friction k de Brunone, plutot que la method traditionnelle par essais et erreurs pour estimer k. Les deux modeles de friction non permanente ont ete incorpores dans un algorithme de calcul du coup de belier par la methode des caracteristiques. Les resultats numeriques obtenus a partir d'un modele de friction quasi permanente et a partir des modeles non permanents de Zielke et de Brunone sont compares avec des resultats de mesures en laboratoire pour des ecoulements laminaires ou turbulents a faible nombre de Reynolds. Des conclusions sont tirees sur les domaines de validite des trois modeles de friction. En complement, la convergence et la stabilite des modeles sont abordees. IntroductionTraditionally the steady or quasi-steady friction terms are incorporated into the standard water hammer algorithms. This assumption is satisfactory for slow transients where the wall shear stress has a quasi-steady behaviour. Experimental validation of steady friction models for rapid transients [1,2,3,4,5] previously has shown significant discrepancies in attenuation and phase shift of pressure traces when the computational results are compared to the results of measurements. The discrepancies are introduced by a difference in ve...

A mixed MOC/FDM numerical formulation for hydraulic transients

“…The results of the numerical example are in good accordance with the available experimental data [6]. Apart from the obvious qualitative agreement, the comparison of quantities, such as the reduction of maximal pressure values in time, is also in accordance with the experiments.…”

“…or according to Vitkovsky's modified formulation [6], which additionally takes into account the correct sign of the convective term, λ is defined by the formula [6,11] …”

“…With the defined pipe and water characteristics the celerity is obtained to be a=1319 m/s. The described pipe system defines the laboratory apparatus used to study water-hammer phenomena at the Robin hydraulics laboratory at the University of Adelaide (for more details see [6]). The valve closure time is set to t c =0,009 s. Within this time interval three different scenarios are numerically simulated: sudden, linear and bilinear reductions of the flow area in the valve.…”

“…The main reason of these disagreements was the inappropriate definition of the fiction factor that in such rapid events should account for unsteady behavior of wall shear stress. To include such effects, several constitutive models were suggested for quantifying the contribution of frictional forces [5,6,7]. The implementation of such constitutive models in a onedimensional flow model was explored for different numerical methods such as: finite difference method (FDM), finite volume method, finite element method, wave plan method and method of characteristics (MOC).…”

Effect of drop in pipe wall stiffness on water-hammer speed and attenuation

“…The wall shear stress τ w is the summation of a quasi-steady and an unsteady friction component. The former is estimated using the Darcy-Weisbach or the Hazen-Williams formulae, whereas the latter results from empirically-based models such as in Bergant et al (2001) or from physically-based models such as in Vardy et al (1993). The wave dissipation due to steady or unsteady friction becomes important for high friction factors or long pipes of small internal diameter (Ghidaoui et al 2005).…”