Analysis of the Francis turbine upper-part-load pulsation Part II – Mechanism of self-excitation

Abstract

When a Francis turbine operates at partial load or very high load, the swirling flow in the draft tube may cause objectionable oscillations of pressure and power. The cavitating core of the vortex plays an important role in these pulsations. The present paper deals with a class of self-excited oscillations of the entire water column in the power plant; self-excitation means that at least one eigenvalue of the hydraulic system becomes unstable. A one-dimensional (1D) model in frequency domain explains how the normal damping is eliminated. Oscillation power is provided in regions whose flow gain in streamwise direction has a component in phase with pressure. The model contains a module for the dynamic transmission behavior of the cavitating vortex; it represents the response of the cavity size to variations of the local pressure and swirl. The sensitivity to pressure changes (the ‘cavitation compliance’) controls the natural frequencies but cannot cause instability whereas the response to swirl changes (‘mass flow gain’) may supply net oscillation energy and thus cause instability. Both influences act all along the cavitating part of the vortex; it is crucial that the variation of runner exit swirl can propagate along the vortex only with the axial velocity of the fluid. The oscillation energy balance depends on the wavelength of swirl variation, i.e. the combination of axial velocity and oscillation frequency. All instabilities of this class are ‘breathing’ pulsations, synchronous within one cross section. In the simplest case with the lowest natural frequency the pressure variation is roughly synchronous in the whole draft tube; for this mode (full-load surge) a lumped-parameter model may be adequate. By contrast, the upper-part-load pulsation occurs in a more complex eigenmode; a distributed-parameter model version is required to represent the essential features. The draft tube pressure oscillation has two quarter waves and a pressure node within the cavitation zone. The pressure at both ends of the draft tube cavitation zone has roughly opposite phase. Difficulties to transpose the stability between reduced-scale model and prototype are explained using the 1D model, as well as some influence of the runner hub shape and of the upstream conduit. Damping at the runner explains why the pulsation is limited to low-head turbines.