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The massive penetration of the electrical network by renewable energy sources, such as wind and solar, pushes the operators to extend hydropower plant units operating range to meet the transmission system operator requirements. However, in off-design operating conditions, flow instabilities are developing in Francis turbines, inducing cavitation, pressure pulsations and potentially resonance that can threaten the stability of the whole system. Reduced scale model testing is commonly performed to assess the hydraulic behaviour of the machine for industrial projects. However, it is not possible to directly transpose pressure pulsations and resonance conditions from model to prototype since the characteristics of the hydraulic circuits are different from model to prototype. In this paper, a methodology developed in the framework of the HYPERBOLE European research project for predicting the eigenfrequencies of hydropower plant units operating in off-design conditions is introduced. It is based on reduced scale model testing and proper one-dimensional modelling of the hydraulic circuits, including the draft tube cavitation flow, at both the model and prototype scales. The hydro-acoustic parameters in the draft tube are identified at the model scale for a wide number of operating conditions and, then, transposed to the full-scale machine, together with the precession frequency for part load conditions. This enables the prediction of the eigenfrequencies and resonance conditions of the full-scale generating unit.

At off-design operating operations, cavitating flow is often observed in hydraulic machines. The presence of a cavitation vortex rope may induce draft tube surge and electrical power swings at part load and full load operations. The stability analysis of these operating conditions requires a numerical pipe model taking into account the complexity of the two-phase flow. Among the hydroacoustic parameters describing the cavitating draft tube flow in the numerical model, the mass flow gain factor, representing the mass excitation source expressed as the rate of change of the cavitation volume as a function of the discharge, remains difficult to model. This paper presents a quasi-static method to estimate the mass flow gain factor in the draft tube for a given cavitation vortex rope volume in the case of a reduced scale physical model of a ν = 0.27 Francis turbine. The methodology is based on an experimental identification of the natural frequency of the test rig hydraulic system for different Thoma numbers. With the identification of the natural frequency, it is possible to model the wave speed, the cavitation compliance and the volume of the cavitation vortex rope. By applying this new methodology for different discharge values, it becomes possible to identify the mass flow gain factor and improve the accuracy of the system stability analysis.

At part load conditions, Francis turbines experience the formation of a cavitation vortex rope at the runner outlet whose precession acts as a pressure excitation source for the hydraulic circuit. This can lead to hydro-acoustic resonances characterized by high pressure pulsations, as well as torque and output power fluctuations. This study highlights the influence of the discharge factor on both the vortex parameters and the pressure excitation source by performing Particle Image Velocimetry (PIV) and pressure measurements. Moreover, it is shown that the occurrence of hydro-acoustic resonances in cavitation conditions mainly depend on the swirl degree of the flow independently of the speed factor. Empirical laws linking both natural and precession frequencies with the operating parameters of the machine are, then, derived, enabling the prediction of resonance conditions on the complete part load operating range of the turbine.

Francis turbines operating in part load conditions experience a swirling flow at the runner outlet leading to the development of a precessing cavitation vortex rope in the draft tube. This cavitation vortex rope changes drastically the velocity of pressure waves traveling in the draft tube and may lead to resonance conditions in the hydraulic circuit. The wave speed being strongly related to the cavitation compliance, this research work presents a simple model to explain how it is affected by variations of operating conditions and proposes a method to transpose its values. Even though the focus of this paper is on transpositions within the same turbine scale, the methodology is also expected to be tested for the model to prototype transposition in the future. Comparisons between measurements and calculations are in good agreement.

Francis turbines are subject to various types of the cavitation flow depending on the operating condition s. In order to compensate for the stochastic nature of renewable energy sources, it is more and more required to extend the operating range of the generating units, from deep part load to full load conditions. In the deep part load condition, the formation of cavitation vortices in the turbine blade-to-blade channel s called inter -blade cavitation vortex is often observed. The understanding of the dynamic characteristics of these inter -blade vortices and their formation mechanisms is of key importance in an effort of developing reliable flow simulation tools. This paper reports the numerical and experimental investigations carried out in order to establish the vortex characteristics, especially the inception and the development of the vortex structure. The unsteady RANS simulation for the multiphase flow is performed with the SST - SA S turbulence model by using the commercial flow solver ANSYS CFX. The simulation results in terms of the vortex structure and the cavitation volume are evaluated by comparing them to the flow visualization s of the blade channel acquired through a specially instrumented guide vane as well as from the downstream of the runner across the draft tube cone. The inter-blade cavitation vortex is successfully captured by the simulation and both numerical and experimental results evidence that the inter -blade vortices are attached to the runner hub.

In real systems, the phenomena, such as pipe-wall viscoelasticity, unsteady friction or fluid structure interaction induce additional damping and dispersion of transient pressure waves than that defined by classical waterhammer. In this paper, unsteady friction models and viscoelastic damping models will be presented and a theoretical formulation of the viscoelastic damping in piping systems without cavitation will be developed. Firstly, the friction factor will be presented as the sum of the quasi-steady part and the unsteady part related to the instantaneous local acceleration and instantaneous convective acceleration. This unsteady friction model has been incorporated into the method of characteristic algorithm (MOC). Secondly, the damping will be defined in terms of viscoelastic effect attributed to a second viscosity µ’. This model is solved using the Finite Difference Method. Finally, numerical results from the unsteady friction and viscoelastic models are compared with results of laboratory measurements for waterhammer cases with low Reynolds number turbulent flows. This comparison validates the new viscoelastic model.