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Abstract
The dynamic characteristics of running away pump turbines (PTs) with a large head variable amplitude have not been understood thus far, primarily because of two difficulties in simulation and analysis. The first is how to provide accurate time-varying boundary conditions for transient simulation of the turbine runaway process (TRP). The other is how to determine the specific appearance time of each frequency component of the complex pressure fluctuations. This study presented a one- and three-dimensional (1D-3D) coupled approach considering waterway dynamics to provide accurate unsteady boundary conditions for the transient flow simulation of a PT with a large head variable amplitude. The short-time Fourier transformation (STFT) approach was adopted to analyse the time-frequency characteristics of the transient pressures and impeller forces. The study found that the fluctuations of pressures and impeller forces during the TRP of the PT with a large head variable amplitude contained two exclusive fluctuation frequency components. The former was approximately three times the rated rotational frequency of the impeller. The later was a series of integer fold transient rotational frequencies of the impeller, which was irrelevant to the rotor-stator interactions. The findings have important value for controlling the pressure fluctuations during the TRP of PTs.
Disclosure statement
No potential conflict of interest was reported by the author(s).
Notations
| A | = | area of pipe across section (m2) |
| a | = | acoustic speed (m/s) |
| CFD | = | computational fluid dynamics |
| D | = | inner diameter of pipe (m) |
| Dt | = | impeller inlet diameter |
| DDES | = | delayed detached eddy simulation |
| E | = | elastic modulus of pipe (Pa) |
| e | = | thickness of pipe wall (m) |
| F | = | impeller hydraulic thrusts (N) |
| f | = | frequency (Hz) |
| fi | = | mass force (m/s2) |
| fBFV | = | frequency of pressure fluctuation related to backflow vortex (Hz) |
| fDV | = | frequency of pressure fluctuation related to DV in spiral casing (Hz) |
| fn | = | rotation frequency of PT impeller (Hz) |
| fRSI-S and fRSI-R | = | frequency of pressure fluctuation related to RSI (Hz) |
| fRFI | = | frequency of pressure fluctuation related to RFI (Hz) |
| FVM | = | finite volume method |
| fw | = | friction coefficient of pipe wall (-) |
| FX | = | impeller force in X direction (N) |
| FY | = | impeller force in Y direction (N) |
| FZ | = | impeller axial force (N) |
| g | = | gravitational acceleration (m/s2) |
| H | = | hydraulic head (m) |
| Ht | = | PT head (m) |
| Hr | = | rated head of investigated PT (m) |
| J | = | Rotor inertia (kg·m2) |
| K | = | elastic modulus of water (Pa) |
| LES | = | large eddy simulation |
| M | = | impeller torque (N·m) |
| M11 | = | unit torque (N·m) |
| MOC | = | method of characteristic |
| n | = | impeller rotational speed (r min−1) |
| n11 | = | unit speed (r min−1) |
| nr | = | rated rotational speed of impeller (r min−1) |
| ns | = | rated specific speed (r/min) |
| P | = | rated power of investigated PT (kW) |
| p | = | pressure (Pa) |
| Q | = | flowrate of the pipe (m3 s−1) |
| RANS | = | Reynolds average Navier-stokes simulation |
| RSI | = | rotor-stator interaction |
| STFT | = | short time Fourier transformation |
| t | = | time (s) |
| u | = | flow velocity component (m/s) |
| V | = | mean flow velocity in pipe across section (m/s) |
| x | = | spatial coordinate (m) |
| α | = | angle of the pipe axis relative to the horizontal plane (°) |
| ν | = | kinematic viscosity (m2/s) |
| ρ | = | water density (kg m−3) |
| φ | = | coefficient related to the elasticity and connection characteristics of the pipe (-) |
| σ | = | Poisson's ratio of pipe wall material (-) |
| π | = | circular constant (-) |
| Δt | = | time-step size (s) |
| Δx | = | spatial step size (m) |
Subscripts
| BFV | = | backflow vortex |
| DT | = | draft-tube inlet |
| DV | = | Dean vortexes |
| X, Y, Z | = | three directions in Cartesian coordinates |
| IM | = | high-pressure inlet of impeller |
| i, j | = | Cartesian coordinate components (i, j = 1, 2, 3) |
| N | = | sequence index of time step |
| r | = | at rated condition |
| RFI | = | periodical interactions between impeller and complex flow patterns in distributor |
| SE | = | spiral casing end |
| t | = | turbine |
| VL | = | vaneless space |
| 0 | = | initial state |