Impact of perturbed eddy-viscosity modeling on stability and shape sensitivity of the hydro-turbine vortex rope using linearized Reynolds-averaged Navier-Stokes equations

Abstract

This study investigates the influence of a perturbed eddy-viscosity model on linear stability and shape sensitivity of the global vortex rope mode arising in a hydro-turbine flow under fully turbulent conditions. The framework is based on the Reynolds-averaged Navier--Stokes equations with a standard $k$-$\varepsilon$ turbulence closure, linearized around a base-flow state. This base state is tuned to match the vortex-rope bifurcation predicted from three-dimensional unsteady simulations. The shape sensitivity of the global mode is derived, accounting for perturbations of both the base flow and the linear operator. We show that although the perturbed eddy-viscosity model has only a marginal effect on the eigenvalues and eigenmodes of interest, it substantially alters the resulting shape sensitivities. These differences arise primarily through the base-flow contribution to the total sensitivity, which dominates the sensitivity to shape deformations. Although both models identify coherent-velocity production and advection as the leading contributors, the linearized model captures additional mechanisms associated with eddy-viscosity perturbations. Comparison with experiments shows that only the perturbed eddy-viscosity model reproduces the correct trends in shape sensitivity, whereas the frozen model fails to do so. These findings highlight the importance of consistently linearizing turbulence models for sensitivity-based control of turbulent global instabilities.

Figures (17)

• Figure 1: Geometrical domain of the RANS simulation and LSA setup.
• Figure 2: Computational mesh of the RANS simulation and LSA.
• Figure 3: Base-flow fields in a streamwise section for (a) axial velocity, (b) radial velocity, (c) azimuthal velocity, and (d) eddy viscosity at $Q^* = 0.65$. Gray dashed line indicates wake half width $y_{1/2}$.
• Figure 4: Eigenvalue spectrum for $m=1$ comparing frozen and perturbed eddy-viscosity model at $Q^* = 0.65$. The respective eigenvalues corresponding to the vortex-rope instability are indicated by the filled markers.
• Figure 5: Spatial mode shapes at arbitrary phase angle in a streamwise section comparing (top) perturbed and (bottom) frozen eddy-viscosity model. Displayed are the eigenmodes of (a) axial velocity, (b) radial velocity, (c) azimuthal velocity, and (d) eddy viscosity at $Q^* = 0.65$. Gray dashed line indicates wake half width $y_{1/2}$.