# Séminaire de Mécanique d'Orsay

## Le Jeudi 21 mars à 14h00 - Salle de conférences du LIMSI

### Extended Squire transformation and its applications to optimal growth analysis

## J John Soundar Jerome

Institut Jean Le Rond D'Alembert, FCIH. UPMC

The Squire transformation is extended to the entire eigenfunction structure of both Orr-Sommerfeld & Squire modes and to the initial value problem in the case of confined parallel shear flows. Then, its implications on the optimal transient growth of arbitrary 3D disturbances is studied. Thereby, large Reynolds number asymptotics for the optimal gain at all optimization time t with t/Re finite or large are derived. The Squire transformation is also extended to the adjoint problem. It is, thus, demonstrated that the long-time optimal growth may be decomposed as a product of the gains arising from purely 2D mechanisms and an analytical contribution representing 3D growth mechanisms but scaling as (Beta.Re)^2, where Beta is the spanwise wavenumber. For example, when the leading eigenmode is an Orr-Sommerfeld mode, it is given by the product of respective gains from the 2D Orr-mechanism and the lift-up mechanism. Direct numerical solutions of the optimal gain for plane Poiseuille and plane Couette flow confirm the novel predictions of the Squire transformation extended to the initial value problem.