Analysis of transient growth using an orthogonal decomposition of the velocity field in the Orr-Sommerfeld Squire equations

TL;DR

This paper introduces an orthogonal decomposition of velocity fields in the Orr-Sommerfeld Squire equations to better understand transient growth and non-orthogonality of eigenmodes in wall-bounded shear flows, revealing new insights into flow transition.

Contribution

It proposes a novel orthogonal decomposition of velocity perturbations, providing a dimensionally homogeneous eigenmode basis and clarifying the non-orthogonality and resonance effects in transient growth analysis.

Findings

Eigenmodes are non-orthogonal, allowing large transient growth.

The new formulation links optimal modes to continuous mode transition.

Numerical results illustrate destabilizing perturbations in boundary layers.

Abstract

Despite remarkable accomplishment, the classical hydrodynamic stability theory fails to predict transition in wall-bounded shear ow. The shortcoming of this modal approach was found 20 years ago and is linked to the non-orthogonality of the eigenmodes of the linearised problem, de noted by the Orr Sommerfeld and Squire equations. The associated eigenmodes of this linearised problem are the normal velocity and the normal vorticity eigenmodes, which are not dimensionally homogeneous quantities. Thus non-orthogonality condition between these two families of eigenmodes have not been clearly demonstrated yet. Using an orthogonal decomposition of solenoidal velocity fields, a velocity perturbation is expressed as an L2 orthogonal sum of an OrrSommerfeld velocity field (function of the perturbation normal velocity) and a Squirevelocity field (function of the perturbation normal vorticity). Using this decomposition,a variational formulation of the linearised problem is written, that is equivalent to the Orr Sommerfeld and Squire equations, but whose eigenmodes consist of two families of velocity eigenmodes (thus dimensionally homogeneous). We demonstrate that these two sets are non-orthogonal and linear combination between them can produce large transient growth. Using this new formulation, the link between optimal mode and continuous mode transition will also be clari fied, as the role of direct resonance. Numerical solutions are presented to illustrate the analysis in the case of thin boundary layers developing between two parallel walls at large Reynolds number. Characterisations of the destabilizing perturbations will be given in that case.