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Publication# Hydroacoustic Modeling of a Cavitation Vortex Rope for a Francis Turbine

Résumé

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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With economical energy market strategies based on instantaneous pricings of electricity as function of the demand or the predictions, operators harness more hydroelectric facilities to off-design operating points to cover the variations of the electricity production. Under these operating conditions, Francis turbines develop a cavitating swirling flow at the runner outlet which induces pressure fluctuations propagating in the whole hydraulic system. The core of this cavitating vortex is usually called vortex rope. At resonance conditions, the superimposition of the induced traveling waves gives rise to a standing wave leading to undesirable large pressure and output power fluctuations. The aim of this present work is to predict and to simulate this resonance phenomenon which may happen both in part load or full load operating conditions. The identification of the excitation sources induced by the cavitating vortex rope is performed with numerical simulations based on a three dimensional incompressible model, so called hydrodynamic (HD) model. The assumption of plane wave propagation in the water passages connected to the turbine is set since low surging frequencies are involved. Hence, propagation of these sources is simulated with a one dimensional compressible model, so called hydroacoustic (HA) model. The HA model covers the entire hydraulic system including the source region corresponding to the draft tube of the Francis turbine whereas the HD model covers only the source region. In this present work, a specific HA draft tube model has been developed. A momentum source modeling the forces induced by the flow acting on the draft tube wall is considered. Moreover, the fluctuating cavitation volume is considered as a mass source. Finally, a thermodynamic damping is introduced to model energy dissipation during a phase change between liquid and gas. Investigations at part load conditions aim to simulate the upper part load resonance phenomenon for which frequency of pressure fluctuations are experienced between 2 and 4 times the runner frequency. Measurements were carried out in the framework of the FLINDT project which is therefore the case study for validation. First of all, HA draft tube model parameters have been derived for the investigated operating point considering both single phase and two phase unsteady simulations with the HD model. An analysis of these parameters is performed and comparison between single phase and two phase simulation results is made. It is shown that the cavitation modeling in the HD model is necessary to find the vortex rope precession frequency which depends on the cavitation amount in the vortex core. However, the volume of vapor is underestimated and a correction factor on the Thoma number is necessary to get a good agreement between experiments and simulation results. Moreover it has been shown that the three dimensional flow in the elbow gives rise to HA sources able to excite the hydraulic system. Intensity of the sources are higher when two phase flow simulations are considered. Before simulating the upper part load resonance phenomenon, a preliminary validation of these HA parameters is performed by simulating a standard part load resonance where the vortex rope precession frequency, near 0.3 times the runner frequency, matches with the first eigenfrequency of the hydraulic system. In out of resonance conditions, maximum of pressure fluctuations amplitudes are experienced in the draft tube cone with an amplitude being 1% of the turbine head. However, when resonance occurs, maximum amplitude of pressure fluctuations reaches up to 7%. A good agreement is obtained with the order of magnitudes found in measurements available in the literature. After this preliminary validation, simulation of the upper part load resonance phenomenon has been tackled. It has been found that the mechanism inducing this phenomenon is related to an undesirable fluctuation of the cavitation volume which frequency can match with an eigenfrequency of the hydraulic system. However, this fluctuation is captured for a Thoma number much higher than the experimental one leading to a cavitation volume very small compared to the experiments. A prototype installation of four 478 MW Francis turbines located in the Canada's province British Columbia, has been chosen as the case study to analyze the full load instability phenomenon. Indeed, this instability occurred on prototype and reduced scale model as well. Hence, experimental measurements have been carried out on the reduced scale model aiming to use experimental data to validate the numerical simulations performed with the HA draft tube model. The mass source defined in this model, is described by a decisive parameter which is the mass flow gain factor. Extensively used in previous works for the analysis of this phenomenon, this parameter is defined to represent the effect of the HA fluctuations of the downstream flow rate to the cavitation volume on the mass source. In this present work, the same formulation is used and has been combined with the introduction of a new parameter: the thermodynamic damping. First of all, these HA parameters have been derived for the different investigated experimental operating points from single phase steady simulations. Then, using these computed parameters, a small perturbation stability analysis in the frequency domain has been carried out to identify the stability of the different operating points. The experimental unstable characteristic frequencies have been found out with this modal analysis. However, this analysis in the frequency domain does not give any information about the amplitude of the pressure fluctuations induced by the instability. Hence, time domain HA simulations have been performed. It has been shown that the using of constant HA draft tube model parameters leads to divergent time domain simulations, whereas nonlinear parameters depending on the pressure variable, lead to a limit cycle of finite amplitude fluctuations. Moreover, nonlinearity of the thermodynamic damping is decisive to reach this limit cycle. Finally, a methodology has been set up to predict the instability of the prototype from the investigations on the reduced scale model. A combination of measurements, numerical simulations and computation of the eigenmodes of the reduced scale model installed on test rig, allows the accurate calibration of the HA draft tube model parameters at the model scale. Finally, transposition of these parameters to the prototype according to similitude laws is applied for the stability analysis of the power plant.

Hydropower represented in 1999 19% of the world electricity production and the absolute production is expected to grow considerably during the next 30 years. Francis turbines play a major role in the hydroelectric production due to their extended range of application. Due to the deregulated energy market, hydroelectric power plants are increasingly subjecting to off design operation, start-up and shutdown and new control strategies. Consequently, the operation of Francis turbine power plants leads to transients phenomena, risk of resonance or instabilities. The understanding of these propagation phenomena is therefore paramount. This work is a contribution to the hydroacoustic modelling of Francis turbine power plants for the investigation of the aforementioned problematic. The first part of the document presents the modelling of the dynamic behavior and the transient analysis of hydroelectric power plants. Therefore, the one-dimensional model of an elementary pipe is derived from the governing equations, i.e. momentum and continuity equations. The use of appropriate numerical schemes leads to a discrete model of the pipe consisting of a T-shaped equivalent electrical circuit. The accuracy in the frequency domain of the discrete model of the pipe is determined by comparison with the analytical solution of the governing equations. The modelling approach is extended to hydraulic components such as valve, surge tanks, surge shaft, air vessels, cavitation development, etc. Then, the modelling of the Francis, Pelton and Kaplan turbines for transient analysis purposes is presented. This modelling is based on the use of the static characteristic of the turbines. The hydraulic components models are implemented in the EPFL software SIMSEN developed for the simulation of electrical installations. After validation of the hydraulic models, transient phenomena in hydroelectric power plants are investigated. It appears that standard separate studies of either the hydraulic or of the electrical part are valid only for design purposes, while full hydroelectric models are necessary for the optimization of turbine speed governors. The second part of the document deals with the modelling and analysis of possible resonance or operating instabilities in Francis turbine power plants. The review of the excitation sources inherent to Francis turbine operations indicates that the draft tube and the rotor-stator interaction pressure fluctuations are of the major concern. As the modelling of part load pressure fluctuations induced by the cavitating vortex rope that develops in the draft tube at low frequencies is well established, the focus is put on higher frequency phenomena such as higher part load pressure fluctuations and rotorstator interactions or full load instabilities. Three hydroacoustic investigations are performed. (i) Pressure fluctuations identified experimentally at higher part load on a reduced scale model Francis turbine are investigated by means of hydroacoustic simulations and high speed flow visualizations. The resonance of the test rig due to the vortex rope excitation is pointed out by the simulation while the special motion and shape of the cavitating vortex rope at the resonance frequency is highlighted by the visualization. A description of the possible excitation mechanisms is proposed. (ii) A pressure and power surge measured on a 4 × 400 MW pumped-storage plant operating at full load is investigated. The modelling of the entire system, including the hydraulic circuit, the rotating inertias and the electrical installation provides an explanation of the phenomenon and the related conditions of apparition. A non-linear model of the full load vortex rope is established and qualitatively validated. (iii) The rotor-stator interactions (RSI) are studied in the case of a reduced scale pump-turbine model. An original modelling approach of this phenomenon based on the flow distribution between the stationnary and the rotating part is presented. The model provides the RSI pressure fluctuation patterns in the vaneless gap and enables to predict standing waves in the spiral case and adduction pipe. The proposed one-dimensional modelling approach enables the simulation, analysis and optimization of the dynamic behavior of hydroelectric power plants. The approach has proven its capability of simulating properly both transient and periodic phenomena. Such investigations can be undertaken at early stages of a project to assess the possible dynamic problems and to select appropriate solutions ensuring the safest and optimal operation of the facility.

When water flows through hydraulic turbomachines, the local pressure can become low enough to vaporize the water and create vapor cavities. This phenomenon is called cavitation. When the cavities collapse, shock waves and liquid jets traveling through the inclusions can erode nearby solid surfaces. The collapse of cavitation bubbles has been extensively investigated in the case of a single bubble in a liquid at rest. However, in the case of hydraulic machines, the bubbles collapse in a flowing liquid subject to strong pressure gradients. The objective of this thesis is thus to investigate the effect of the pressure gradient on the collapse of a spherical cavitation bubble. We have performed a preliminary investigation of bubble dynamics in a flowing liquid with a pressure gradient. To this end, we have placed a Naca0009 hydrofoil in the test section of EPFL High Speed Cavitation Tunnel and used a high energy pulsed laser focusing technique to generate a single vapor bubble close to the hydrofoil’s leading edge. We have observed a significant influence of the pressure gradient on the bubble dynamics. Particularly, if the collapse phase occurs near the minimum pressure point, the microjet is no more directed towards the solid surface but towards the lower pressure zone in the stream wise direction. We have also observed a peculiar feature of a cluster of bubble dynamics, which behaves almost similar to a single bubble, exhibiting a microjet during its collapse. These unprecedented re- sults are of major importance for a better understanding of the cavitation erosion mechanism in hydraulic systems. The qualitative results obtained in the cavitation tunnel led to the investigation of the effect of a constant pressure gradient on the collapse of the bubble. An experimental setup is built to observe the dynamics of the bubble in water, subject to the gravity induced hydrostatic pressure gradient, and to measure the pressure fluctuation due to the shock waves. The bubbles are generated with a high energy pulsed laser and recorded with a high speed camera. The experimental setup is taken onboard parabolic flights. The parabolic manoeuvres allow the gravity level to be varied in the plane, which modulates the intensity of the pressure gradient in the liquid. The high speed movies taken during the flights reveal that vapor jets appear with the rebound bubble. An empirical law for the prediction of the volume of the jet is deducted from the experimental results. The volume of the jet, normalized with the volume of the rebound bubble, is found to be proportional to a non dimensional parameter ζ0=|∇p|Rmax/Δp, where ∇p is the pressure gradient, Rmax is the maximal bubble radius and Δp is the driving pressure. This dependance is enforced by a theoretical development based on the concept of the Kelvin impulse. Moreover, we identify a threshold for the apparition of the vapor jet: ζ > 4 · 10 − 4 . A new approach for the study of the bubble collapse is proposed: we look at how the energy in the initial cavitation bubble is partitioned between the collapse channels, namely the rebound, the shock wave, the jet, and the luminescence. The microgravity phases of the parabolic flights prevent the apparition of the jet. The collapse of the bubble, in this case, is perfectly spherically symmetric. Moreover, the energy dissipated through luminescence is negligible. Therefore, the study reduces to the energy partition between rebound and shock wave. The measurements uncover a systematic pressure dependence of the energy partition between rebound and shock. We demonstrate that these observations agree with a physical model relying on a first-order approximation of the liquid compressibility and an adiabatic treatment of the non-condensable gas inside the bubble. Using this model, we find that the energy partition between rebound and shock is dictated by a single non-dimensional parameter ξ=Δp(γ^6)/(pg0^1γ)/(ρc^2)^(1−1/γ), where γ is the adiabatic index of the non-condensable gas, pg0 is the pressure of the non-condensable gas at the maximal bubble radius, ρ is the liquid density, and c is the speed of sound in the liquid.