Vortex breakdown in a hydro turbine draft tube swirling jet

TL;DR

This paper investigates vortex breakdown in a hydro turbine draft tube, revealing bifurcation phenomena and hysteresis effects in swirling flows through simplified laminar flow models, with implications for turbine stability.

Contribution

It introduces a simplified laminar flow model to analyze vortex rope formation and bifurcations in turbine draft tubes, highlighting new insights into flow stability and hysteresis.

Findings

Vortex rope mode bifurcates supercritically from axisymmetric flow.

Hysteresis loop exists due to subcritical solutions.

Transcritical bifurcation occurs at finite Reynolds number.

Abstract

The swirling flow in a Francis type hydropower turbine is known to be susceptible to the formation of a large helical structure, commonly referred to as a vortex rope. This vortex rope can be interpreted as an unstable mode associated with vortex breakdown. This perspective is adopted here in a simplified laminar flow setting. The helical vortex rope mode is shown to bifurcate supercritically from an axisymmetric baseflow in a Hopf bifurcation within a turbine draft tube. When wall friction effects are neglected, a large recirculation region at the axis can form and a range of subcritical solutions is identified for a flow regime corresponding to partial load of the turbine. The existence of these subcritical solutions promotes the emergence of a hysteresis loop. We further describe a regular dynamics of a formation of recirculation bubble at the axis and its destruction due to the emergence of a helical vortex rope at its periphery. Increasing the axial flow discharge towards the regime corresponding to nominal turbine load leads to an unfolding of the steady solutions branch in a transcritical bifurcation. This bifurcation takes place at finite Reynolds number and complements existing evidence of transcritical bifurcation of the swirling jet flows, previously reported only in the inviscid limit.