A review of one-dimensional unsteady friction models for transient pipe flow
Abstract. This paper reviews a quasi-steady model and four unsteady friction models for transient pipe flow. One of the factors which may affect the accuracy of the one-dimensional models of transition flow is the friction coefficient. This coefficient can be estimated as steady, quasi-steady, and unsteady. In the steady approach, a constant value of the Darcy-Weisbach friction factor is used. In the quasi-steady approximation, friction losses are estimated by using formula derived for steady-state flow conditions. The fundamental assumption in this approximation is that the head loss during transient conditions is equal to the head loss obtained for steady uniform flow with an average velocity equal to the instantaneous transient velocity. During transient conditions the shear stress at the wall is not in phase with the mean velocity. In addition, the velocity profile can be completely different from a uniform flow profile. Therefore friction losses computed by using steady-state relationships are inaccurate in transient laminar and turbulent flow. To cope with this problem, for both laminar and turbulent flows, it is possible to algebraically add unsteady-flow terms to the quasi-steady resistance term of one-dimensional models. Unsteady models are divided into two groups. The first group includes those models which instantaneous wall shear stress is the sum of the quasi-steady value plus a term in which certain weights are given to the past velocity changes. Three models of this group are presented in this paper: Zielke, Vardy & Brown, and Trikha. The second group of models assumes the wall shear stress due to ﬂow unsteadiness is proportional to the variable ﬂow acceleration. Brunone model from this group is presented in this paper. Numerical results from the quasi-steady friction model and the Zielke, Vardy & Brown, Trikha 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. The computational results clearly show that Zielke model yields better conformance with the experimental data
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