Modal analysis of a spinning disk in a dense fluid as a model for high head hydraulic turbines

Abstract

In high head Francis turbines and pump-turbines in particular, Rotor Stator Interaction (RSI) is an unavoidable source of excitation that needs to be predicted accurately. Precise knowledge of turbine dynamic characteristics, notably the variation of the rotor natural frequencies with rotation speed and added mass of the surrounding water, is essential to assess potential resonance and resulting amplification of vibrations. In these machines, the disk-like structures of the runner crown and band as well as the head cover and bottom ring give rise to the emergence of diametrical modes and a mode split phenomenon for which no efficient prediction method exists to date. Fully coupled Fluid-Structure Interaction (FSI) methods are too computationally expensive; hence, we seek a simplified modelling tool for the design and the expected-life prediction of these turbines.We present the development of both an analytical modal analysis based on the assumed mode approach and potential flow theory, and a modal force Computational Fluid Dynamics (CFD) approach for rotating disks in dense fluid. Both methods accurately predict the natural frequency split as well as the natural frequency drift within 7.9% of the values measured experimentally. The analytical model explains how mode split and drift are respectively caused by linear and quadratic dependence of the added mass with relative circumferential velocity between flexural waves and fluid rotation.

A Review of the Efficiency Improvement of Hydraulic Turbines in Energy Recovery

“…Louyot proposed an analytical modal analysis method based on an assumed modal approach and potential flow theory and a modal force computational fluid dynamics (CFD) method. Both methods accurately predict intrinsic frequency splitting and intrinsic frequency drift with errors within the experimentally allowed limits [37][38][39][40].…”

Analysis of the Mode Shapes of Kaplan Runners

“…In Table 1 some mode shapes of a disk clamped at its center are shown and sorted by the number of nodal diameters (D) and nodal circles (C). As it was studied by many authors [17,26,[28][29][30][31], the vibration mode shapes of a disk have combinations of nodal diameters and nodal circles, which increase their number with the mode order. For convenience, in this work, the mode shapes related to the disk will be called modes "(m, n)" for modes shapes with m nodal diameter and n nodal circles.…”

“…It was found that while frequencies of the disk decrease when the disk gets close to the rigid wall, the opposite occurs when the disk gets close to the non-rigid wall. Louyot et al [17] made a modal analysis of a spinning disk in a dense fluid as a model for a high-head hydraulic runner. The mode shapes were extracted and, an expression for co-and counter-rotating wave angular frequency was analytically determined.…”

Assessment of Novel Modal Testing Methods for Structures Rotating in Water

“…For instance, the ND vibration modes evolve from standing mode shapes under non-rotating conditions to traveling mode shapes under rotating conditions. Under non-rotating conditions, the displacements of each point of the disk are either in phase or out of phase, while under rotating conditions the displacements are phase-shifted, resulting in a wave traveling over the disk [17,[20][21][22].…”