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

TL;DR

This study analyzes the linear stability of a turbulent swirling jet at a Francis turbine draft-tube outlet using local LSA with three eddy-viscosity closures. It demonstrates that turbulent diffusion markedly damps growth rates and confines unstable azimuthal modes to $m\in[-1,2]$, with partial-load (0.92 BEP) as the most unstable regime, while WKB and inviscid criteria offer complementary insight. Adjoint-based sensitivity reveals a localized inner-core wavemaker region, showing axial baseflow strongly influences growth rate and azimuthal baseflow governs frequency, and highlights the necessity of spatially varying $\nu_t$ to predict the unstable mode range. The approach is fast and modular but neglects draft-tube geometry, motivating future global stability analyses and sensitivity-informed flow-control strategies.

Abstract

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.

Figures (31)

• Figure 1: (a) Sketch of a Francis turbine. The draft tube is located downstream of the turbine runner. (b) Three-dimensional visualization of the velocity distribution at the inlet of the draft tube mean swirling flow profile for $0.92$ BEP. The azimuthal $U_\theta$ and axial $U_z$ velocity distributions are shown in blue and red arrows, respectively. The vortex core radii ($R_1$ and $R_2$) associated with the primary swirling regions, are shown as dashed circles.
• Figure 2: Baseflow profiles of the three turbine operating points. (a) Azimuthal ($U_{\theta}$, solid line) and axial ($U_z$, dashed line) velocities, and (b) the corresponding turbulent kinetic energy ($K$) profiles.
• Figure 3: Turbulent viscosity ($\nu_t$) profiles for three turbine operating points. The profiles correspond to the constant model ($\overline{\nu_t}$, dotted lines), the mixing-length model ($\nu_{t,\mathrm{ml}}$, dashed lines), and the $k\!-\!\varepsilon$ model ($\nu_{t,k\!-\!\varepsilon}$, solid lines).
• Figure 4: Growth rates $\lambda$ as functions of the axial wavenumber $k$, with each branch corresponding to a different azimuthal wavenumber $m$. The figure is organized in a $4 \times 3$ grid: rows represent different turbulence models: (a) $\nu_t = 0$, (b) $\overline{\nu_{t}}$, (c) $\nu_{t, \mathrm{ml}}$, and (d) $\nu_{t, k-\varepsilon}$, while the columns correspond to operating conditions: (1) $0.92$ BEP, (2) BEP, and (3) $1.06$ BEP. The dashed lines correspond to the predicted maximum growth rate defined in \ref{['eq:max_lambda']}.
• Figure 5: Leibovich & Stewartson instability criteria defined in \ref{['eq:condition']} for the different baseflow conditions. The region in which $\tilde{\varPhi} < 0$ at 0.92 BEP is highlighted in yellow.