Extended Squire's transformation and its consequences for transient growth in a confined shear flow

TL;DR

This paper extends Squire's transformation to eigenfunctions of Orr-Sommerfeld and Squire modes, enabling analysis of 3D disturbance growth in shear flows through 2D problems, revealing scaling laws and mechanisms for transient growth.

Contribution

The authors extend Squire's transformation to the entire eigenfunction structure and adjoint modes, linking 3D transient growth to 2D mechanisms and deriving new scaling laws.

Findings

Predicts large-Reynolds-number scaling for optimal gain at all times.

Decomposes 3D growth into 2D mechanisms and an analytical 3D contribution.

Shows lift-up mechanism dominates in Squire modes.

Abstract

The classical Squire transformation is extended to the entire eigenfunction structure of both Orr-Sommerfeld and Squire modes. For arbitrary Reynolds numbers Re, this transformation allows the solution of the initial-value problem for an arbitrary three-dimensional (3D) disturbance via a two-dimensional (2D) initial-value problem at a smaller Reynolds number Re2D. Its implications for the transient growth of arbitrary 3D disturbances is studied. Using the Squire transformation, the general solution of the initial-value problem is shown to predict large-Reynolds-number scaling for the optimal gain at all optimization times t with t/Re finite or large. This result is an extension of the well-known scaling laws first obtained by Gustavsson (J. Fluid Mech., vol. 224, 1991, pp. 241-260) and Reddy & Henningson (J. Fluid Mech., vol. 252, 1993, pp. 209-238) for arbitrary \alpha Re, where \alpha is the streamwise wavenumber. The Squire transformation is also extended to the adjoint problem and, hence, the adjoint Orr-Sommerfeld and Squire modes. It is, thus, demonstrated that the long-time optimal growth of 3D perturbations as given by the exponential growth (or decay) of the leading eigenmode times an extra gain representing its receptivity, may be decomposed as a product of the gains arising from purely 2D mechanisms and an analytical contribution representing 3D growth mechanisms equal to 1+(\beta Re/Re2D)2G, where \beta is the spanwise wavenumber and G is a known expression. 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 an analytical expression representing the 3D lift-up mechanism. Whereas if the leading eigenmode is a Squire mode, the extra gain is shown to be solely due to the 3D lift-up mechanism.