ASYMPTOTIC MULTIPLE-SCALES ANALYSIS OF HYDRAULIC TRANSIENTS IN ELASTIC AND VISCOELASTIC PRESSURIZED CONDUITS
The hydraulic transient phenomenon known as water hammer has a long history . 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. , Chapter 4), water hammer is treated for variable valve-closure time using a momentum equation extended to partially developed turbulence (Hansen et al.  1995,  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.  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. , Chapter 5), an unsteady friction model introduced by Brunone et al.  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. ) were used here. The analytic form surprisingly predicts that the steady viscous extension (Hansen et al.  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. , 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  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. ). This contribution includes: (i) resolution of an outstanding paradox (Weinerowska-Bords  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.