ASYMPTOTIC MULTIPLE-SCALES ANALYSIS OF HYDRAULIC TRANSIENTS IN ELASTIC AND VISCOELASTIC PRESSURIZED CONDUITS

Abstract

The hydraulic transient phenomenon known as water hammer has a long history [35]. To date, only relatively simple cases have been studied analytically among the numerous publications. In this research, a formal asymptotic wave attenuation form is found for each of three water hammer models, i.e., the classic model, an unsteady friction model, and a generalized Kelvin-Voight model. Explicit dependence of pressure-wave attenuation on lumped parameters is found and a deeper and more direct understanding of the water hammer phenomenon is obtained that is unavailable from numerical methods. Firstly, (Yao et al. [99], Chapter 4), water hammer is treated for variable valve-closure time using a momentum equation extended to partially developed turbulence (Hansen et al. [43] 1995, [42] 2011). A closed-form pressure-wave attenuation, found via multiple-scales asymptotics, is useful over much longer times than previous ad hoc results (Tijsseling et al. [52] 2000). Numerical validation of analytical results is obtained using the method of characteristics and convergence verified through a time and space grid reductions. Experimental validation was not considered. The contribution includes: (i) understanding of parametric dependence of pressure-wave attenuation, (ii) capacity to handle time-varying closure cases which is currently unavailable, and (iii) validity in case of flow reversals given uniform fluid movement. Secondly, (Yao et al. [98], Chapter 5), an unsteady friction model introduced by Brunone et al. [16] 1990 to account for turbulence, is considered. An extension of Chapter 4 is used to find the wave attenuation form. Increased attenuation due to unsteady friction is reduced to a single time-dependent exponential factor involving the product of Brunone’s unsteady-friction parameters. The numerical solution, found as before, was used to verify analytic results. Model parameters chosen to match experimental results (Bergant et al. [8]) were used here. The analytic form surprisingly predicts that the steady viscous extension (Hansen et al. [42] 2011), accounting for partially developed turbulence, provides an equally viable explanation for the increased pressure-wave attenuation given a weak spatial dependence. Finally, (Yao et al. [100], Chapter 6), the multiple-scales method (Chapters 4 and 5) is further extended to water hammer in viscoelastic pipe modelled using a Kelvin-Voight representation. Pipe plasticity is found to increase the pressure wave attenuation rate via a third time scale driven by weak strain-rate feedback. A Preissmann weighted fourpoint box-scheme (Weinerowska-Bords [90] 2006) is used to obtain numerical solutions to the mathematical model. The analytic work is validated by matching to a numerical solution matched to experimental data (Mitosec et al. [59]). This contribution includes: (i) resolution of an outstanding paradox (Weinerowska-Bords [90] 2006) involving an order of magnitude mismatch between predicted Kelvin-Voight parameters and those required to match numerical and experimental data, and (ii) introduction of a novel classification method for the prediction of the scale of lumped-parameters without access to experimental data. Both (i) and (ii) predict that this work will be useful in the planning and design phases of experiment and field installations.