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

TL;DR

This paper examines how perturbing eddy-viscosity models affects the stability and shape sensitivity analysis of vortex rope modes in hydro-turbine flows, emphasizing the importance of consistent turbulence model linearization.

Contribution

It demonstrates that linearizing turbulence models significantly influences shape sensitivity results and improves agreement with experimental trends in turbulent flow stability analysis.

Findings

Perturbed eddy-viscosity model alters shape sensitivities substantially.

Consistent linearization of turbulence models is crucial for accurate sensitivity analysis.

Only the perturbed model reproduces experimental shape sensitivity trends.

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.