Linear Stability and Structural Sensitivity of a Swirling Jet in a Francis Turbine Draft Tube

Motivated by the need to better understand flow unsteadiness in hydraulic turbines, we perform a local linear stability and adjoint-based sensitivity analysis of the turbulent swirling jet at the outlet of a Francis turbine. We use measured mean flow and turbulence profiles at several operating conditions (below, at, and above the best efficiency point (BEP) flow rate) and perform a stability analysis. Incorporating eddy viscosity $ν_t$ into the analysis strongly damps inviscid growth rates and restricts instability to low azimuthal modes $m\in [-1,2]$, in better agreement with experiments. Three turbulent viscosity closures (constant, mixing-length and measured $k-\varepsilon$ based) yield similar spectra, with close agreement between mixing length and measured models, all identify partial load (0.92 BEP) as the most unstable regime. Sensitivity results show that axial velocity modifications primarily control growth rates, whereas azimuthal velocity changes mainly shift frequencies. We also derive the sensitivity kernel of the spectrum to turbulent viscosity modifications and find that spatial variations of eddy viscosity are essential for predicting the unstable mode range. The predictions accurately estimate stability changes for small variations in operating point. We further analyze the flow using classical inviscid swirling jet instability criteria (the generalized Rayleigh discriminant) and WKB analysis to predict the stability to broader operating points and reconcile these results to the stability and sensitivity analyses. The approach used in this study is fast and simple to model, but it neglects draft tube geometry (non-parallel effects), motivating future global stability and sensitivity analyses.

💡 Research Summary

This paper presents a comprehensive investigation into the flow instabilities occurring in the draft tube of Francis turbines, specifically focusing on the swirling vortex rope phenomenon under off-design operating conditions. Motivated by the need for operational flexibility in hydropower plants integrating renewable energy, the study employs local linear stability and adjoint-based sensitivity analysis on the turbulent swirling jet at the turbine outlet.

The analysis is grounded in experimentally measured mean flow and turbulence profiles at several operating points: below (0.92 BEP), at (BEP), and above (1.06 BEP) the best efficiency point. The base flow is modeled as a superposition of three vortices (rigid-body rotation and two Batchelor vortices) for the azimuthal velocity and a freestream with two Gaussian jets for the axial velocity. A key innovation is the incorporation of turbulent effects via a “frozen eddy viscosity” approach into the linear stability analysis. Three closure models for the eddy viscosity are compared: a constant value, a mixing-length model, and a model derived from measured turbulent kinetic energy (k) and dissipation rate (ε).

The results demonstrate that including eddy viscosity strongly damps the inviscid growth rates and confines instability to low azimuthal wavenumbers (m ∈