Hydroacoustic Modeling of a Cavitation Vortex Rope for a Francis Turbine

Hydraulic machines subject to off-design operation involve the presence of cavitating flow regimes in the draft tube. The cavitation vortex rope at part load conditions is described as an excitation source for the hydraulic system, and interactions between this excitation source and system eigenfrequency may result in resonance phenomena and induce a draft tube surge and electrical power swings. To accurately predict and simulate a part load resonance, proper modeling of the draft tube is critical. The presence of this cavitation vortex rope requires a numerical pipe element taking into account the complexity of the two-phase flow. Among the parameters describing the numerical model of the cavitating draft tube flow, three hydroacoustic parameters require a special attention. The first hydroacoustic parameter is called cavitation compliance. This dynamic parameter represents the variation of the cavitation volume with respect to a variation of pressure and implicitly defines the local wave speed in the draft tube. The second parameter corresponds to the bulk viscosity and is related to internal processes breaking a thermodynamic equilibrium between the cavitation volume and the surrounding liquid. The third parameter is the excitation source induced by the precessing vortex rope. The methodology to identify these hydroacoustic parameters is based on the direct link that exists between the natural frequency of the hydraulic system and the wave speed in the draft tube. First, the natural frequency is identified with the help of an external excitation system. Then, the wave speed is determined thanks to an accurate numerical model of the experimental hydraulic system. By applying this identification procedure for different values of Thoma number, it is possible to quantify the cavitation compliance and the void fraction of the cavitation vortex rope. In order to determine the energy dissipation induced by the cavitation volume, the experimental hydraulic system is excited at the natural frequency. With a Pressure-Time method, the amount of excitation energy is quantified and is injected into the numerical model. A spectral analysis of the forced harmonic response is used to identify the bulk viscosity and the pressure source induced by vortex rope precession. Thus, the identification of the hydroacoustic parameters requires the development of a new numerical draft tube model taking into account the divergent geometry and the convective terms of the momentum equation. Different numerical draft tube models are compared to determine the impact of convective and divergent geometry terms on identification of the hydroacoustic parameters. Furthermore, to predict the hydroacoustic parameters for non-studied operating conditions and to break free from the dependence upon the level setting of the Francis turbine, dimensionless numbers are proposed. They have the advantage of being independent from the selected numerical model and they define a behavior law of hydroacoustic parameters when the cavitation volume oscillates at resonance operating conditions. Finally, to investigate the stability operation of the prototype, the hydroacoustic parameters need to be transposed to the prototype conditions according to transposition laws. By assuming both Thoma similitude and Froude similitude conditions, transposition laws are developed and the hydroacoustic parameters are predicted for the prototype.

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