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Abstract
Quasi two dimensional (2D) and one dimensional (1D) numerical simulations of transient flow with column separation are conducted using the discrete vaporous and gaseous cavitation models (DVCM and DGCM) in a simple reservoir-pipeline-valve system. In the quasi 2D transient model, the governing equations are solved by the method of characteristics along the pipe and by finite difference across the pipe cross section to consider the velocity profile, and the final model is coupled with the cavitating models. The comparison between the numerical and experimental results shows that the quasi 2D DGCM is more successful in predicting the maximum head pressures and reproducing the pressure spikes at different water hammer cycles. The quasi 2D model of DGCM correlates better with the experimental data than the quasi 2D DVCM, 1D DVCM and 1D DGCM in terms of pressure magnitude. In sensitivity analysis, the 2D DGCM shows more grid convergence than the 1D models of DGCM and DVCM and 2D DVCM by increasing the number of longitudinal and radial intervals.
Disclosure statement
No potential conflict of interest was reported by the authors.
Notation
| A | = | cross-sectional flow area (m2) |
| Ca, Cb, Cc, Cm | = | boundary layer coefficients (−) |
| C* | = | Vardy’s shear decay coefficient (−) |
| c | = | liquid wave speed (m s−1) |
| D | = | internal pipe diameter (m) |
| f | = | Darcy–Weisbach friction factor (−) |
| fq | = | quasi-steady friction (−) |
| fu | = | unsteady friction (−) |
| g | = | gravitational acceleration (m s−2) |
| H | = | pressure head (m) |
| hv | = | vapour pressure head (m) |
| k | = | Brunone’s friction coefficient (−) |
| L | = | pipe length (m) |
| Nx, Nr | = | numbers of computational sections (−) |
| Q | = | discharge (l s−1) |
| Qu, Qd | = | upstream and downstream discharges of computational sections (l s−1) |
| q | = | radial velocity (m s−1) |
| R | = | radius of pipe (m) |
| Re | = | Reynolds number (−) |
| Re* | = | Reynolds number based on u* (−) |
| r | = | distance from the axis in radial direction (m) |
| t | = | time (s) |
| u | = | local longitudinal velocity (m s−1) |
| uu, ud | = | upstream and downstream local longitudinal velocity (m s−1) |
| u* | = | friction velocity (m s−1) |
| u′, v′ | = | velocity fluctuation in axial and radial direction (m s−1) |
| V | = | average velocity (m s−1) |
| v | = | local radial velocity (m s−1) |
| x | = | distance along the pipe (m) |
| y | = | radial distance from wall (m) |
| y* | = | dimensionless wall distance (−) |
| Δt | = | time step (s) |
| Δx | = | reach length (m) |
| κ | = | von Karman’s constant (−) |
| υ | = | kinematic viscosity (m2 s−1) |
| υt | = | eddy viscosity (m2 s−1) |
| ρ | = | density of liquid (kg m−3) |
| ψ | = | weighting factor (−) |
| τ, τw | = | shear stress (kg m−1 s−2) |
| = | volume of vapour/gas cavities (m3) | |
| = | volume of vapour/gas mixture (m3) |