Vortex breakdown in a hydro turbine draft tube swirling jet

TL;DR

This work analyzes vortex rope formation in a Francis turbine draft-tube by treating the rope as a vortex-breakdown instability of a swirl-dominated, axisymmetric baseflow in a laminar setting. Using linear stability analysis and direct numerical simulations on a simplified diffuser–pipe domain, it identifies a supercritical Hopf bifurcation of the $m=1$ mode at $Re_c \approx 2340.86$ with dominant frequency $ω \approx 1.262$, leading to a saturated monohelical vortex rope whose amplitude scales as $\sqrt{Re-Re_c}$. Boundary conditions markedly alter the bifurcation: no-slip walls give the classical Hopf scenario, while free-slip walls yield axis-centered recirculation, two saddle-node folds and a breathing, hysteretic dynamics, with an imperfect transcritical unfolding observed as inflow swirl (BEP) changes. The results illuminate how vortex rope-like structures emerge from vortex breakdown in turbine geometries and point to future work incorporating turbulence models to bridge toward realistic turbulent rope dynamics.

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.

Figures (20)

• Figure 1: Domain composed of a conical diffuser and a straight cylindrical section susan2010analysis. Flow enters the domain on the left and exists on the right. $R_1=D_1/2=1.063$, $R_2=D_2/2=1.6$, $L_1=3.58$, $L_2=15$.
• Figure 2: Inlet velocity profile corresponding to the 0.92 and 0.98 best efficiency point (black solid and blue dashed line respectively). Parameter values are specified in table \ref{['tab1']}. Radial velocity $U_r=0$. Additional exponential regularisation is applied near the wall at $r=1.063$ as discussed in the text.
• Figure 3: Azimuthal component of the baseflow (colorscale) for $Re=500$ (a), $Re=1000$ (b), $Re=2000$ (c) and $Re=3000$ (d). Overlapped isocontour of streamfunction $\psi$ ranges from 0 to 0.16 with increment 0.02 (solid line), equals $\psi_1=0.162$ (dashed line), and ranges from $\psi_1+10^{-3}$ to the maximal $\psi$ value with increment $10^{-3}$ (dotted line).
• Figure 4: Spectrum of the linearised Navier-Stokes operator for the azimuthal wavenumber $m=1$ (a). Subset of 40 eigenvalues closest to the shift $\sigma=-1.1i$ is shown. The critical Reynolds number is $Re_c\approx2340.86$. Real part of axial velocity $\hat{u}_z$ of three eigenvectors of leading growth rate at $Re=3000$ is shown in the panels (b-d). Amplitude of the eigenvector is determined by the chosen normalisation.
• Figure 5: Time series of perturbation energy observable \ref{['enpert']} upon Nek5000 time integration of the steady state solution obtained with FreeFem++. The observable remains at a low residual level for $Re<Re_c$ and saturates nonlinearly for $Re>Re_c$. Rate of the exponential growth is compared with the results from linear stability analysis for $Re=2400$ and $2500$.