[論文レビュー] Impact of perturbed eddy-viscosity modeling on stability and shape sensitivity of the hydro-turbine vortex rope using linearized Reynolds-averaged Navier-Stokes equations

Jens Müller, Sophie Knechtel+1 more|arXiv (Cornell University)|Mar 4, 2026

Fluid Dynamics and Turbulent Flows被引用数 0

ひとこと要約

tldr: The paper analyzes how perturbed eddy-viscosity modeling affects linear stability and shape sensitivity of the hydro-turbine vortex rope, showing substantial changes in shape sensitivities while eigenvalues are only marginally affected, and validates against experiments.

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.

研究の動機と目的

• Motivate and quantify the impact of turbulence-model perturbations on linear stability of the vortex rope in a fully turbulent hydro-turbine flow.
• Develop a rigorous shape-sensitivity framework that accounts for base-flow and operator perturbations in a RANS context.
• Compare frozen and perturbed eddy-viscosity models to identify missing mechanisms and validate against experimental trends.
• Provide decomposition of shape sensitivity into feed-back and base-flow contributions and link to physical mechanisms.

提案手法

• Base the analysis on unsteady RANS with a standard k–epsilon closure and linearize about a turbulence-based base flow.
• Derive linearized RANS equations including perturbations of eddy viscosity and turbulent kinetic energy.
• Formulate a normal-mode stability problem with azimuthal symmetry and solve a discretized eigenvalue problem.
• Compute adjoint eigenmodes and derive shape sensitivities using a discrete shape-deformation approach.
• Decompose shape sensitivity into feed-back and base-flow contributions, and further into physical mechanisms via the Hessian-based terms.
• Contrast perturbed versus frozen eddy-viscosity models and relate findings to experimental trends.

実験結果

リサーチクエスチョン

• RQ1 What is the influence of the perturbed eddy-viscosity model on eigenvalues and eigenmodes of the vortex-rope instability?
• RQ2 How does perturbing eddy viscosity alter shape sensitivities compared to a frozen model?
• RQ3 What base-flow or mechanism contributions dominate the shape sensitivity, and how do these compare with experimental observations?
• RQ4 Which physical mechanisms are captured by the perturbed model that are neglected by the frozen model?

主な発見

• The perturbed eddy-viscosity model has only a marginal effect on the eigenvalues and eigenmodes of interest.
• The perturbed model substantially alters the resulting shape sensitivities, with base-flow contributions dominating the total sensitivity.
• Both models identify coherent-velocity production and advection as leading contributors to sensitivity, but the perturbed model reveals additional mechanisms tied to 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.